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By reading others questions like Highest damage in one melee attack and Highest AC possible.
How many hit points can a character have, and also highest achievable hit points possible in an instant?
Conditions:
In terms of permanent HP, you can't get better than a level 20 Hill Dwarf Barbarian with the following features:
...and the following magic items:
This totals to 465 (or 560 with maximum rolls on hit dice):
145 from hit dice (240 with maximum rolls)
200 from Constitution
20 from Dwarven Toughness
40 from the Tough feat
40 from the Boon of Fortitude
20 from the berserker's axe
By using a luck blade to cast wish and replicating the effect of the following spells1:
...our barbarian can gain additional hit points.
Heroes' feast provides between 2 and 20 HP (an average of 11)
Aid provides either 35 or 40 HP (depending on your interpretation of whether wish replicates up to 8th level or 9th level effects; see this Q&A)
All together, this yields 502-525 for average results on hit dice, and 597-620 for maximum results on hit dice.
1 credit to Medix2 in the comments
The most temporary hit points that can be achieved is by using luck blade once more, this time to replicate mass polymorph. The spell allows you to retain your normal hit points in the new form but gain temporary hit points equal to the HP of the new form.
The best beast form to assume is the huge giant crab found in the "White Plume Mountain" module of Tales from the Yawning Portal for 161 temporary hit points. Alternatively, if your GM rules that you need to choose a standard form, you can choose the giant ape for 157 temporary hit points.
With:
That gives us:
\begin{align} &12\text{ (per level from hit die)} \\+{}&10 \text{ (per level from CON)} \\+{}&\,\,\, 2 \text{ (per level from Tough feat) } \\+{}&\,\,\, 1 \text{ (per level from Dwarven Toughness) } \\+{}&\,\,\, 1 \text{ (per level from Berserker's Axe) } \\={}& (12+10+2+1+1) \times 20 \\={}& 520 \end{align}
One of the Epic Boons (DMG, p. 232) is the Boon of Fortitude, which says:
Your hit point maximum increases by 40.
So if we add that to the above, that becomes 560 HP.
1 It's possible to get stats to 30 using a tome multiple times. However, usually that requires magic to be around long enough for it to regain its magic after a century has passed.
A dwarf could use the tome three times in their life without the aid of magic due to being able to live up to 350+ years, so long as they became a level 20 Barbarian (so that their CON can be 24 before using the Manual of Bodily Health) really young, before they turned 50.
Either that or (since the PHB says they live 350 years on average) this dwarf was just exceptional and lived up to, say, 360 years (and became a level 20 barbarian between the ages of 50-60). Either way, if they could fit in reading the tome three times after becoming a level 20 barbarian, they could have a CON of 30.
Or a simpler answer, this barbarian was able to get hold of multiple Manual of Bodily Health...
align environment work for you?
\$\endgroup\$
A level 20 Circle of the Moon Druid has an unlimited number of Wildshape uses available thanks to its Archdruid feature:
At 20th level, you can use your Wild Shape an unlimited number of times.
When specialized into the Circle of the Moon, she also picks up Combat Wild Shape:
When you choose this circle at 2nd level, you gain the ability to use Wild Shape on your turn as a bonus action, rather than as an action.
This allows her to shift out of one form and into a new one on the same turn, resetting her form's HP. Because a level 20 Circle of the moon Druid has Elemental Wild Shape, she can choose an Earth Elemental with 126 HP and resistance to mundane attacks to provide a lot of "padding" for damage before she resets her form the next turn with either an action or bonus action.
From a functional standpoint, this might be considered "healing" to some folks, not infinite HP, but unlike with healing, no spell slots are used nor is concentration required.
Note: The question Can a druid wildshape again whilst still in animal form? discusses the fact that a Druid can wildshape into a new form while already wildshaped. It does not discuss the possibility of a Druid wildshaping into its current form (ie "refreshing" the form).
If the table ruling is that the Druid must choose a new form, not the current form, the Druid could switch back and forth between a Mammoth and Earth Elemental (both have 126 HP) or use both an action and bonus action to end and then re-Wild Shape to maintain only the Earth Elemental.
Using a 20th level single class character with feats and with magic items from the DMG, we can get 425 hit points.
We use the barbarian for the d12 hit die. This gives 145 hit points at level 20.
We need at least Con 12 from the start. We use the bonus from hill dwarf for Con+2 and three ASI to get to 20 Con. With more starting Con we can use less ASI for this.
At level 20 the barbarian's Con increases by 4 to 24. We can then use multiple manuals of bodily health to increase the value to 30(+10) Alternatively, this is possible using the optional advancement rules for after level 20.
With the fourth ASI we get the tough feat for +2 hit points per level. Further, we use the berserker axe and the hill dwarf subclass for a further +1 hit point per level each.
We get 145 points from hit dice, 200 points from constitution and 80 points from the other bonuses for a total of 425 points.
Rolling the hit dice can give up to 95 additional hit points.
This is assuming we get a 12 every time. This would increase the hit points to 520.
Boon of Fortitude can give 40 more points.
This is an optional advancement for after level 20. See DMG p. 232. It would take the hit points to 465 or 560, respectively.
At level 20, we can use true polymorph to turn into a monster permanently with a CR of 20 or less. For hit point optimization, the best monster from the MM that can be chosen is the dragon turtle with CR 17 and 341 hit points.
As a 20th level bard, with True Polymorph, Hero's Feast, and Aid among your Magical Secrets, assemble an appropriate hoard of resources, cast a few spells into Glyphs of Warding (using scrolls where appropriate), then True Polymorph yourself into a dragon turtle (CR 17). That gives you 341 HP on the average numbers, or 450 on a max roll. Following that, spend the time to properly read and make use of 5 Manuals of Bodily Health, increasing your constitution (as a Dragon Turtle) to 30, and thus giving you an extra 5 HP per hit die - taking you to 451/560.
We can do a bit better than that if we don't limit ourselves to the Monster Manual, but it seemed that the question was about core books.
Boon of Fortitude, also post-dragon-turtle, gives us another 40 HP, taking us to 491/600. This is complicated by the fact that Epic Boons are only available to 20th level characters. Is a 20th level wizard who's been polymorphed into a Dragon Turtle still a 20th level character?
Aid cast at the 9th level from a Glyph of Warding gives you 45 hp. Heroes Feast from another Glyph of Warding gives you another 11.
Absolute max before temp HP: 656
If temp HP are included, it's just a matter of adding in a lvl 9 Armor of Agathys (again, Magical Secrets, Glyph of Warding, and scroll crafting are your friends) to crank the total up to 701.
There's a fair amount of cheese in getting to that 701, though.
