Your damage does not increase by level. (1)
(1) At least not directly. You get more feats, hit more often, have higher base stats, may aquire magic items and get stronger spells. But the raw damage does not increase. If a weapon does 1d6 damage, it does so, regardless of your level. Exceptions will be noted in the damage text.
You can make this Goat part of your world's Folklore
What you can do is create a new monster, per DMG p. 273-283, with a twist.
What you are doing in this case is, since you are interested in this goat moving up as the PC moves up, is creating an "evolving" monster. You can grow the goat in the same way dragons increase in size and power as they age. (He'll do it faster, of course). That is one template. Since the goat is adventuring, though, it would be worth folding in elements of Creating a new race/species (p. 285 DMG) of Goat or Giant Goat.
Heck, let him grow! Paul Bunyon's ox Babe was larger than average! The trick is to find the sweet spot in the ratio of CR to the player character's level. (see below). As this is homebrew, it may be an iterative process.
Use the point value guide in the MM for various abilities and skills that you give to the goat to bump up its CR as the character goes up in level.
What CR should I shoot for? Good question, as the match up from CR to PC Lvl is not one-for-one.
Consider the Archmage NPC who can cast level 9 spells, and who has a CR of 12, and then consider that a CR of 12 is a suitable challenge of 3 or 4 characters at level 12 ... it all depends on the skills, hit dice, and abilities you give to your goat.
As it levels up, add some features to your goat to bump its CR up:
- HD
Bump up ASI. Consider giving it boosts to INT to where it can
nearly talk/communicate via baa's and grunts with the PC.
Damage per attack
- Increasingly effective "knock prone" or "Knock back" melee
abilities.
- Number of attacks per round.
- Increased perception, be it visual, olfactory, audio, taste.
- Increase resistances to X.
- Increase dexterity (ASI)
- Increase Constitution (ASI)
- Increase Wisdom
When you upgrade the goat's kit/package every few levels, look at all of the features you now have and try to match it to a CR as if it were a monster. If the CR is about 1/4 to 1/3 of the PC's level, you should be in the ball park of not getting it to be too powerful, but still has a chance to survive adventures as the character goes up in level. When your PC is about 18th level, a CR between 5 and 6 ought to fit well enough. That puts him between a Triceritops (10d12, CR 5) and a Mammoth (11d12, CR6) in challenge rating, though he need not be that huge if you don't want him to be. Some of his HD increase can reflect how he's learned to handle danger/combat better.
At some point, this goat will become famous, as it is the companion of a hero, and will be like Babe the Blue Ox. You can decide/rule that the goat has had enough adventuring, and either wants to be put out to stud, put out to pasture, or wants to go to sea. (Every ship has a goat locker where the salty old NCO's and Bosun's mates hang out. A perfect place for a wise and experienced old goat).
Consider the stud fees you can charge for this amazing goat ...
Experience with something like this: it's been a few years, but I had to create an evolving PC/Monster when the dwarf in the campaign I was running, in 1e, died and got reincarnated as a Brass dragon. When that roll came up, I was puzzled at how I'd keep the player interested in continuing on. I was fortunate: the age/hd template was already there, but it was tricky (and we had a few false starts) in keeping Miles from being too powerful compared to the party. As the group never got past 8th level, we never got to see him blossom, but it was instructive nonetheless. 5e has a lot more guidance on how to pull this off.
Best Answer
The crux of the question revolves around what chasing a high AC actually accomplishes. Without knowing the end goal, it's impossible to judge whether or not you've fulfilled your aims. This answer is going to proceed under the assumption that you're just trying not to get physically hit the majority of the time rather than completely mitigate your opponent's offense. The latter is better accomplished by utilizing miss chances and conditions.
A good metric for "relevant" would be your average opponent needing an 11 or better to hit you, leaving aside circumstantial effects like flanking, cover, etc. A "high" AC would be closer to needing a 15+ to hit you, so we'll use both of these as our guideposts.
Numeric Analysis
A survey of all the CR 1 monsters in the SRD yields an average attack bonus of +2.7 with two outliers at +5 (small air/earth elemental) and +6 (grig). Using our above targets, that suggests an AC of 14 to 18, with an absolute cap of 21 for the ultra-cautious. The vast majority sit right at +2 to +3, so shooting for average is perfectly safe at this level.
At CR 5, the average attack bonus rises to +9.7, with high outliers at +13 (Greater Barghest and Werebear hybrid/bear form). This suggests an AC of 21 to 25 with a cap at 28.
Moving on to CR 10, the average attack bonus is now +18.6, with two high outliers at +25 (Colossal Animated Object and Juvenile Red Dragon). Target ACs should be 30 to 35 with a cap of 40. At this point, the divide between attach-oriented monsters and special-ability-oriented monsters begins to get wildly apparent. The high attacks are more than 12 points higher than the low attacks. Raw AC is not likely to be the single answer it was at lower levels.
At CR 15, the field is almost exclusively dragons, which skews the average attack bonus to +26.9. Expanding to include CR 14 monsters drops the average to +24.8. The outliers are pretty much every dragon in the book at +28 to +33. Average AC target jumps to 36 to 40 with a cap of 48. A realistic gauge of attacks is hard to establish because chances are you're facing something with class levels, templates, or special abilities that don't rely on attack rolls, such as spells.
At CR 20, you're down to very powerful dragons, top-tier NPCs, pit fiends, and balors. Average attack bonus is +37.0, but swings from +30 to +46, excluding whatever overpowered BBEG your GM has in mind. AC targets are difficult to properly set due to such a small sample size, but 50 to 55 is a decent ballpark. At this stage, raw AC doesn't mean as much because you'll be facing numerous save-or-die effects, massive area effects, and other non-attack offensive abilities. Even a cheap ring of evasion will probably save your life more than cranking out that last 10 to 15 AC points.
Getting There
Actually achieving any of these AC targets is an entirely different kettle of fish.
At 1st level, a decent suit of armor or an exceptional Dexterity score will get you to AC 14 to 16 easily. Add on a heavy shield and you're at AC 18 without even trying.
At 5th level, it starts requiring some effort. To hit an AC of 25:
That even leaves you enough gold for a +1 weapon and a few other trinkets.
At 10th level, you're going to have to invest a significant portion of your wealth (approx. 45,000 gp) to hit your AC target of 34-39:
Above 10th, conventional AC adds are going to eventually top you out at 57 for a total cost of about 237,000 gp (half that if you can find a caster to sink about 9120 XP into crafting it):
The earliest you could manage this would be between levels 15 and 16 if you're buying/acquiring them or 13 to 14th if you're having them crafted.
With some creativity in feat selection and finding a way to add various other stats to your AC (Int and Wis are the most obvious), you can decrease your reliance on items somewhat, but that's left as an exercise for the reader.