As of the 10th printing, the minimum gained is 1 hit point.

The 10th printing (based on the errata published in November 2018) now indicates that the minimum gained at level up is 1 hit point. See this answer for more details.

The following is the original text of this answer, left as a historical footnote (or a way to justify the minimum of 1 prior to the errata).

There is no lower bound rule for hit points gained at level-up, but one of the designers seems to intend it.

Discussion on this lack of a rule has gone back several years. There are claims that a line appeared in a version of the D&D Next playtest (the precursor to 5th edition) that put a lower limit, but I can find no evidence of this publicly available. If there was such a rule, it did not make it into the Player's Handbook, and there has been no errata or Sage Advice to address the issue either. The by-the-book answer to your question is that there is no such rule.

It appears that one of the game's designers, Mike Mearls, believes there should be a 1 hit point minimum gained at each level-up (see this tweet in which he says the "minimum is 1"), but that is neither a rule nor an official ruling, since it was not included in the errata or in Sage Advice or stated by Jeremy Crawford, whose tweets can be considered authoritative.

However, if one of the rules designers believes such a rule is implicit, then it's probably safe to assume that's the way the game is intended to be played even if the statement was not from an (officially) authoritative source. If you are okay with an as-intended interpretation of the rules, it appears that you gain at least 1 hit point at each level-up.

Expected Average Damage: 641.354, Maximum Damage: 1185

No items or spell components, no special circumstances. Legal for use in Adventurer's League. Attacking with Surprise.

Half-Orc (Arthur) (Savage Attacks add an extra die of damage on every critical hit)

Divine Soul Sorcerer 4 (Favored by the Gods, Shadow Blade spell, Subtle Spell meta-magic, ASI)

War Domain Cleric 1 (War Priest - make one weapon attack as a Bonus Action)

Assassin Rogue 3 (Assassinate - if surprised, advantage on all attacks, critical on all hits, 2d6 Sneak Attack with Finesse weapons per turn, Expertise: Stealth, Perception)

Battle Master Fighter 3 (Action Surge, 4 Maneuvers, including Precision Attack, Menacing Attack, and Disarming Attack, Fighting Style Dueling)

Paladin 2 (Divine Smite, Divine Sense, Fighting Style Armored)

Divination Tradition Wizard 2 (Portent)

Gloom Stalker Ranger 5 (Dread Ambusher: in the first round you gain an extra attack on every Attack action in the first round, +10 movement, extra 1d8 damage of the weapon's type on this extra attack, Primeval Awareness, Pass without a Trace, Longstrider, ASI, Fighting Style Archery)

Strength 14 + 2 + 2 ASI

Dexterity 13

Constitution 8 + 1

Intelligence 13

Wisdom 13

Charisma 13

• To-Hit: +11, +12-19 with Precision Attack
• Hit Points: 92
• AC: 19 (Breastplate, Armored Fighting Style, Shield)
• Stealth: +23 (Expertise and temporary boost from Pass without a Trace)
• Initiative: +2 (+1 Dexterity, +1 Wisdom from Dread Ambusher)
• Perception: +13 (Expertise)
• Level 10 spellcaster (4 Sorcerer + 1 Cleric + (2 Paladin + 5 Ranger)/2 + 2 Wizard)
• Movement speed: 40 feet (50 feet with Longstrider)
• Darkvision: 90ft

The crux of it is that you have created a truly dreadful ambush character which has significant precision-boosting skills and tracking skills for his enemies. Beyond just being able to hit things hard, they also are able to do so with uncanny accuracy. This character goes patrolling for evil and predators in the woods, using his Primeval Awareness (L1 spell slot) to identify that there is evil in his land. As he uses his hunting ability to stalk his prey, his high perception and natural stealth make it difficult for his quarry to hide and make himself similarly difficult if not nearly impossible to spot. He'd use Pass Without a Trace, perhaps subtly, so that he could sneak up on his quarry. As he draws near and identifies his target, he casts Longstrider from his Ranger spells using the Subtle Spell meta-magic just to give himself that extra boost in movement that he'll need in order to close with his quarry before it realizes the danger. He knows how long it will take him to get into striking range, and stealthily moves to within 50 feet of his target. He has foreseen this day's events, and knows that death awaits his enemy. Patiently, he waits until it's his turn to strike. In terms of the sequence of events, I imagine it would look like this:

Primeval Awareness of the quarry, Subtle Pass without a Trace + Subtle Longstrider during the hunt for the quary. In the preparation round, cast Subtle Shadow Blade L5, from which the surprise round begins. Portent determines the initiative order for the target and the attacker: + Assassinating Critical: Action Surge + Dueling Attacks + Divine Smites + Dread Ambusher + Sneak Attack + Bonus Action War Priest weapon attack with Divine Smite.

Resource Expenditure:

• Spending 2L5, 3L4, 3L3, Spell Slots, all maneuvers, action surge, divine favor, War Priest ability
• Optional for flavor Spend another L2, 2L1 Spell slots: use a level 1 spell slot for Primeval Awareness to identify that your enemy exists nearby, then cast Pass without a Trace and Longstrider using Subtle Spell to hunt them and ambush them successfully.

Order of Events:

Preparation Round: Level 5 Shadow Blade. (4d8 psychic damage per melee attack)

Surprise Round: Initiative: Use Portent rolls to guarantee Assassinate feature works. Action: Attack action (2 attacks from Ranger, +1 attack from Dread Ambusher feature) *Bonus Action: 1 attack through War Priest Action Surge: Attack action (2 attacks from Ranger, +1 attack from Dread Ambusher feature)

The Math on the Damage:

(All dice are doubled because these are critical hits because of Assassinate)

Every attack that hits results in this: 2\$\times\$4d8 Psychic (Shadow Blade, Level 5) + 1d8 Savage Attack + 2 Dueling Fighting Style + 5 Strength modifier = 9d8 + 7 Psychic damage

7 attacks (3 attacks per action, plus 1 attack from War Priest) yield: 63d8 + 49 Psychic damage

Gloom Stalker Ranger allows us to adds 1d8 damage of the original weapon's damage type to the extra attack of every Attack action we take in the first round of combat through the Gloom Stalker feature. Since we take two Attack actions through Action Surge, we get extra 2d8 Psychic damage, which is doubled since they're critical hits. 67d8 + 49 Psychic damage

Assassin Rogue allows us to add 2d6 Sneak Attack damage because Shadow Blade is a Finesse weapon. Sneak attack rules for rogues do not require you to use Dexterity when using Finesse weapons. Since the attack is a critical hit, the damage gets doubled. 67d8 + 4d6 + 49 Psychic

As a paladin, every successful hit allows us to add 2d8 + (spell slot level-1)\$\times\$1d8 radiant damage, maxing out at 5d8 for an L4 spell slot or higher. As a 10th level spellcaster, after casting Shadow Blade at level 5, we can use 3 L4 spell slots, 1 L5 spell slot, and 3 L3 spell slots. All of this, of course, gets doubled on critical hits, which these will be. This results in the following additional damage: 2\$\times\$(3\$\times\$5d8 + 1\$\times\$5d8 + 3\$\times\$4d8) = 64d8 Radiant damage

Damage Total:

(67d8 + 4d6 + 49 Psychic) + (64d8 Radiant) = (67\$\times\$4.5 + 4\$\times\$3.5 + 49 Psychic) + (64\$\times\$4.5 Radiant) = 364.5 Psychic + 288 Radiant = 652.5 total

In order for the Assassinate feature to work, during the first round of combat the Assassin must act before their quarry. By having access to Portent, we will have two dice rolls which can be used for any ability check, which includes initiative. Because the quarry has a +0 to their initiative and our character has a +2 to initiative, regardless of what we roll, we can substitute the lower of the two Portent rolls to the target and use the higher of the two rolls for ourselves.

Damage, adjusted for accuracy:

The following probability calculations below were made with the help of the table in the article "D&D 5e: Probabilities for Advantage and Disadvantage" to figure out the likelihood of rolling particular numbers on a 20-sided die with advantage.

At least one attack will always hit. It's true that it's mathematically possible for a 1 to be rolled on every single attack die (7 attacks, all with advantage), but the odds of that are ridiculously low: \$\left(\left(\frac{1}{20}\right)^2\right)^7\$ = 0.0000000000000000006103515625. In other words, the Sneak Attack will always be added.

Precision Attack maneuvers and the Favored by the Gods feature allow the player to use them after the attack roll is made and any of these three abilities can be used in conjunction with one another.

With our target having an AC of 20, and our character having a starting +11 to hit, we need to roll 9 or higher in order to successfully land a hit, which from the table we can see that we have an 84% likelihood through advantage.

The Precision Attack maneuver from the Battle Master, adds 1d8 to attack rolls, yielding 4.5 average increase in likelihood. This can be used on 4 attacks, resulting in +15.5 to attack rolls, since we have 4 superiority dice from the Battle Master. In addition, the Favored by the Gods Divine Soul Sorcerer feature will allow us to add 2d4 to one attack that misses, turning it into a hit on average, due to effectively adding +5 to hit (resulting in a +16 to hit) for one of the two attacks which we cannot add the Battle Master die to. As a result, for 1 attack we need to roll 4 or higher through advantage to score a hit, and for 4 of the other 5 attacks we need to roll 4.5 or higher through advantage in order to score a hit.

The optimal strategy of their use should look something like this:

1. Our attacks, whenever possible, should be sequenced from most damaging to least damaging, in order to take advantage of our ability to lazily add attack bonuses.
2. Similarly, we should use our spell slots for Divine Smite in a decreasing progression. The Bonus Action attack will use the L5 spell slot, the first three attacks of the first Attack action will each use an L4 spell slot, and the three attacks of the second Attack action will each use an L3 spell slot.
3. If an attack misses and we still have a superiority die, we should use the Precision Attack adds an average of 4.5 to the attack roll.
4. If the total for the attack roll is still too low and we still can use Favored by the Gods, we should use it.
5. If the number of attacks remaining is less than the number of superiority dice remaining and we haven't applied a maneuver to the attack, we should apply a damaging maneuver like Menacing Attack or Disarming Attack.

The Bonus Action weapon attack will average (2\$\times\$4d8 + 1d8 + 7) Psychic + 2\$\times\$5d8 Radiant = 47.5 Psychic + 45 Radiant = 92.5 damage. Adjusted for the likelihood of a natural 1, the accuracy adjustment reduces the expected damage total by: -92.5*(1-0.84) = -14.8

When using Favored by the Gods, our to hit becomes +16 on average, which means we need to roll 4 or higher through advantage. On the referenced table, we see that for this we have a 97.8% likelihood. The damage for this attack will be (2\$\times\$4d8 + 1d8 + 7) Psychic damage and (2\$\times\$5d8) Radiant = 9d8 + 7 Psychic + 10d8 Radiant = 47.5 Psychic + 45 Radiant = 92.5 damage. Adjusted for accuracy, this means that the damage total is lessened by -(1-0.978)\$\times\$92.5 = -2.035.

When using the Precision Attack maneuver, since the AC of the target is 20 and we will need to roll "5.5", the average (I think) will be (97.8% + 96%)/2 = 96.9%. This means that each of the 4 attacks with maneuvers has a 96.9% chance to hit. So, we will on average do 96.9% of the expected damage for those 4 attacks. The damage of these 4 attacks of the two Attack actions is: 4\$\times\$1d8 + 2\$\times\$4\$\times\$4d8 + 2\$\times\$1d8 + 4\$\times\$7 Psychic + 2\$\times\$(4d8 + 4d8 + 5d8 + 5d8) Radiant = (38d8 + 28) Psychic + (36d8 Radiant) = 199 Psychic + 162 Radiant = 361 total damage. Adjusting for accuracy, this means that the previously mentioned "total damage" is also lessened by -361\$\times\$ (1 - 0.969) = -11.191

On each of the 4 attacks on which we use a superiority die, if the attack doesn't need the accuracy boost, we can apply the die to the damage by using a different maneuver, such as Disarming Attack, doubling it because of the critical hit. This results in 2\$\times\$1d8 damage = 9 damage. How often would we get to add this extra damage? Since the original to-hit modifier is +11, hitting AC 20 would happen 84% of the time. Thus, 84% of the time, we can substitute a damaging maneuver instead of the Precision Attack Battle Master maneuver. Thus, the expected damage is increased by: 0.84\$\times\$9 onto every attack, for four attacks: 0.84\$\times\$9\$\times\$4 = +30.24

The 3rd attack from the second Attack can do (2\$\times\$4d8 + 1d8 + 7) Psychic damage and (2\$\times\$4d8) Radiant = 9d8 + 7 Psychic + 8d8 Radiant = 47.5 Psychic + 36 Radiant = 83.5 damage. As such, the damage expected for the last attack will be 83.5*(1-0.84) = -13.36.

Result:

Adjusted for accuracy, then, this means that the expected average damage of this character is 652.5 -14.8 - 2.035 -11.191 + 30.24 -13.36 = 641.354 total damage

Hope this math was relatively easy to follow, and hope you guys enjoyed the character and story setup as well. And be aware, the optimal use of the superiority dice and Favored by the Gods could improve this by a few more points.

*Note: Originally, I came up with a slightly higher number by using a Quickened Booming Blade. However, that spell is from Sword Coast Adventurer's Guide, and because many of the other features referenced here are from Xanathar's Guide to Everything, in Adventurer's League we wouldn't be able to use both for this character. *

It's worth noting, since the original poster mentioned this looking like a Death Knight, that this build would probably work even better against a Death Knight, because the latter's shield and sword can be removed through the Disarming Attack maneuver, thus lowering his AC and removing its ability to parry with his reaction.

*Note: Another poster, Herohersh (https://rpg.stackexchange.com/a/138689/49041), cleverly suggested that based on the guidelines for this math problem you could use your action to Create Bonfire during the setup round when you're casting Shadow Blade, step into the bonfire in order to trigger Absorb Elements through your reaction, and then proceed to attack using the bonus damage of Absorb Elements. In addition to probably not working mechanically due to stipulations regarding casting any other spell during the same round during which a Bonus Action spell is cast, for my personal taste I find this unsatisfying because it breaks the immersion and idea that you could still be hidden from your enemy the round before you attack if you conjure a bonfire that gives away your position. *