[RPG] Can Pun-Pun ascend in 5e

You've misinterpreted the movement penalty for grappling and the movement restrictions for jumping. They limit different things.

From the PHB, page 195:

When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.

So you're not halving your jump distance, you're halving your movement speed. And you have to use that for jumping (PHB page 182):

Either way, each foot you clear on the jump costs a foot of movement.

This means that, for a player whose normal movement speed is 30ft, while grappling he can move 15ft, and that's also the maximum jump height he can achieve. Not even the Jump spell or the Boots of Striding and Springing will let him jump further than that. The effects, however, do let him get around the Standing High Jump penalty.

To summarize, when looking into how high you can possibly jump while grappling, you make the calculations in a few steps:

1. Determine your movement speed. This is also your maximum jump height (distance).
2. Halve that height if you're grappling a creature that's not two or more sizes smaller than you.
3. Determine your base jump distance. Use your total Strength to calculate this (including any bonuses to Strength from items, spells or potions).
4. Multiply the base jump distance depending on spells an items you have that increase your jump distance (like the Jump spell).
5. If you're making a Standing High Jump, halve the distance from step 4. Otherwise, subtract 10ft from the height in step 2.
6. Take the lower number between the numbers that you get in steps 2 and 4 (after modifications in step 5). This number is how high you can jump (and how much movement you'll have to spend to jump the full height).

You can use this little tool I whipped up to test out various combinations.

If you find the maximum jump height (step 2) to be restricting you more than your jump distance achievable (step 4), you can use spells and items that increase your movement speed to propel you further. The Haste spell is good to consider. While it is a substantially higher level spell, it lets you use an extra action to Dash, quadrupling your speed in total. The added AC is a nice touch, too. You can even combine it with the Jump spell, as the latter does not require Concentration.

Using just the effects in your example, however, the highest you can jump is 15ft, using a Standing High Jump. If you somehow throw in a Dash, you can jump twice as much. Note that, if the PC's base Strength is already 18, the Belt does not help increase the jump height in this case.

Keep in mind, though, one litle bit about the High Jump rules:

In some circumstances, your DM might allow you to make a Strength (Athletics) check to jump higher than you normally can.

The rule enables you as a DM to let your player jump beyond his maximum movement speed (this bit of the rule is written directly after the bit about the movement speed restriction). When can that happen, how much of a bonus does the player get, what the DC is and what the possible consequences are (chance of Exhaustion, for example) - all of this is up to you.

A commentary on the Belt of Storm Giant Strength

I personally dislike the use of this item. First of all, it is a legendary item, suitable only for the most glorious of adventurers. Giving this item to a player character that's not very high in level (18 at least, I'd say) would seem overpowered, and frankly uninteresting. Even at that level it might not be as good as it sounds. Surely, it's a fantastic item, but it makes all the precious Ability points a grappler had spent on Strength a waste. On the other hand, having the belt in possession early and spending the points somewhere else (or getting feats), the grappler's main mechanic would now depend entirely on the belt. Losing the belt would thus make the character almost useless. While you might not plan on having it taken away from him in any encounter, the player might become paranoid about it nonetheless.

Alternative to using the belt, using a Haste spell (3rd lvl) and the Boots of Striding and Springing (uncommon item), the player can reach 21ft with a Strength of 18, and 24ft with a strength of 20 (assuming a base movement speed of 30ft, and making a Dash with the Haste extra action) by making a normal High jump (with a 10ft running start). Throw in a Jump spell (1st lvl) and the High jump goes up to 50ft. Using these three effects, the Belt would only give an improvement of 4ft with a Standing high jump.

If the Haste spell is not available, and using Dash is not an option, then a better combo would be to use Boots of Speed (rare item) and the Jump spell. Just those two bring the High jump up to 20ft, which might be enough. In this case, too, increasing Strength above 18 brings no benefits (though I'm sure it does so in the other aspects of grappling).

Seeing as all the Belts of Giant Strength do provide great bonuses to almost all aspects of grappling, I would award the legendary version at lvl 20 as a special Grappler capstone. Of course, not without effort. It's a nice excuse to lead your players into a perilous dungeon. I would also house-rule that the belt require a Strength of 20 to wear, just so all those Ability points don't feel like they're going to waste. If your player left his strength at 18, though, that might not be necessary.

At lower levels, a potion of Giant Strength (whichever tier you fancy) might be a suitable reward for a heroic deed, and since it's only temporary it wouldn't break anything (including fun).

With 36 hours to prepare, a level 20 wizard of the School of Necromancy can flip 1392 non-illusory birds in 3 rounds.

The answer provided by NautArch is pretty much unbeatable given the question as posed, but let's assume your mother-in-law has truesight and will therefore be thoroughly unimpressed by any illusory birds you flip at her. Let's see how many real (or at least non-illusory) hands we can make.

Here's the plan:

Let's start the day before the big event. First, you, a level 20 wizard of the School of Necromancy¹, cast simulacrum on yourself, making a copy that is missing only a single 7th-level spell slot. Then you use your other 7th level slot to cast magnificent mansion, inside which you will serve dinner tomorrow. The mansion supplies you with 100 servants whose appearance you decide. Let's assume your imagination is limited to only creatures you know to exist, so you decide to make each servant look like a marelith with 6 hands (because if there's no limit then this answer ceases to be interesting).

With all your remaining 3rd and higher level spell slots, you head over to the local cemetery and cast animate dead. The locals may initially object, but I'm sure that once you explain how awful your mother-in-law is, they'll understand your need. Make sure to emphasize that you're only borrowing their ancestors' sacred remains, and they can have the corpses back tomorrow. Remember to check each corpse for 2 intact middle fingers before animating it! Once this is complete, you cast Leomund's tiny hut as a ritual in the cemetery and camp for the night with your simulacrum, taking a long rest.

In the morning, you awake refreshed and take stock of the 78 animated undead you created yesterday. Then, you continue the work you began last evening, using all your 3rd through 6th level spell slots and your 8th level spell slot, as well as all your simulacrum's 3rd through 8th level spell slots, to animate an additional 138 undead. (Did I mention you should do this in large city with an ample supply of corpses?) With all 216 animated corpses in tow, you return to your mansion to cast a 7th level conjure animals into a 7th-level glyph of warding. Your advance preparations are now complete. (Presumably the 100 servants in the mansion can handle the preparation of the meal itself, leaving you to focus on the important matters.)

When your mother-in-law arrives, you and your simulacrum each cast mage hand, using the hands to present your beautifully plated dish (and also ensuring you don't need to waste an action later casting mage hand once the action starts). When she takes a bite and makes her rude comment, your glyph triggers, summoning 24 flying monkeys.² At the same time, you and your simulacrum each cast conjure animals at 9th level (using both your concentration), bringing the total number of flying monkeys to 88. Then, you and your simulacrum each use your next actions to move your mage hands into position, and your next actions after that to cast chill touch, and finally use both of each of your own hands to flip the bird at her (free actions on your turns), completing the springing of the trap over the course of 3 rounds (or 2 rounds for a Wizard 17/Sorcerer 3 with Quickened Spell).

So, let's get the final count of birds flipped:

• 100 near-transparent marelith-resembling servants from the mansion, 6 hands each: 600
• 216 animated undead humanoids, each with 2 hands: 432
• 88 conjured flying monkeys, each with 2 hands and 2 feet capable of flipping the bird: 352
• you and your simulacrum, each with 2 hands and 2 bird-flipping cantrips: 8

Add these all up, and the total number of non-illusory hands (and feet) awaiting your bird-flipping command is 1392. For the sake of completeness, note that you will also be dealing up to 8d8 necrotic damage to your mother-in-law with the two chill touch spells, which adds some injury to this insult.

Since you still have your 1st and 2nd level spell slots left over, you and your simulacrum may as well spend the 4th round casting suggestion on your mother in law, suggesting that she cave to peer pressure and flip herself the bird. If she fails the saving throw, you can add 2 hands to the total, but this isn't guaranteed. Similarly, you might be able to use Charm Person to convince some of the locals to join the party.

With all of these monkeys and undead in your mansion, it's a good thing it will cease to exist in a few hours, because you do not want to clean up the aftermath of this event. Remember that you control the floor plan of the mansion. Make sure to design a dining room that can comfortably accommodate 369 creatures (although a high ceiling should be sufficient to accommodate the flying monkeys).

Oh, and don't forget to tell all the undead afterward to report back to the town for re-burial. Sure, they're still animated, but I'm sure the villagers can sort it all out.

¹Credit to user ttolluz for adding in the School of Necromancy, whose Undead Thralls feature adds one additional undead humanoid per casting of Animate Dead.

²I don't have the exact stats for a flying monkey available, but I'm willing to make some assumptions about their ability to flip the bird.