PokéBase - Pokémon Q&A
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Not much to be said, right? Moves like Counter and Mirror Coat count. Level up only, not breeding or TM and HM.

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Do you want this for only Gen 8, or all Gens?
Shoot need to edit.
Sorry. Forgot to mention that.
This would take hours upon hours to answer. You’d have to look through every single movepool of every single Pokémon in every generation. If you take a minute per Pokémon, which is probably way too fast, it’d give you a grand total of over 14 hours to find a definite answer.
It would be something Like Ditto and Smeargle, which never learn damaging moves by level up, unless the asker wants Pokemon that don't learn damaging moves excluded. :P
Lmao that's right Staka. I totally forgot.
I feel like you could feasibly do this if you single out all the pokemon that don't start out with a damage dealing move and start there, which I am doing now

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This may not be 100% accurate, but after some time searching I found the most interesting cases regarding level up learnsets. First you must know that learnsets have been designed differently through the generations. But all Pokemon must have at least one move programmed at level 1. Second, here's a list of Pokémon that do not learn any damaging move at all, they're kind of obvious anyway

Metapod, Kakuna, Silcoon, Cascoon, Spewpa - Cocoon-like "harden" Pokémon

Abra, Cosmog, Cosmoem - Teleport Pokémon.

Plus there's Ditto and Smeargle, but it's debatable if transform and sketch counts indirectly.

Now, beside those, come these cases. Only on Diamond, Pearl, the trio of Uxie, Azelf and Mesprit learn Confusion at level 21. After that generation they all have damaging moves at level 1.

Then comes Pyukumyuku, who learns Bide / Counter at level 20. This is the top who has a consistent learnset across generations (VII - VIII).

Last relevant mention goes to the most classic candidates of them all, Magikarp and Feebas; both learn Tackle at level 15, after lots of splashing about.

Next come all Pokémon I found without a level 1 damaging move, so they learn one some levels after. There may be a few missing but nothing beating the previous ones listed (Species / Level / Damaging move)

Hoppip 10/8 Tackle (II - V / VI +)
Plusle 10/7 Quick attack (III / VI)
Minun 10/7 Quick arrack (III / VI only)
Burmy 10 Tackle
Azurill 10 Bubble (IV - BW)
Shaymin 10 Magical Leaf
Igglybuff 9 Pound (5 from ORAS - VII)
Spoink 7 Psywave
Ralts 6/4 Confusion (III - VI / ORAS)

It is also worth mentioning that if you count out Metronome as a damaging move, then Togepi would be the absolute latest bloomer regarding damage, learning Ancient Power around level 33. It learns Metronome much earlier than that though (level 7).

Now if you're interested in learnsets in general, they were designed differently across generations, but it's safe to say that after Gen III, the designs are very similar to the present day. Only some exceptions specifically designed have no damaging moves at level 1, or don't learn damaging moves at all.

In generation I, all learnsets were designed before breeding was allowed, so some high leveled Pokémon did not have a significant learnset before their encounter level, i.e. fossil, legendaries, hitmon___ (all one-time only Pokémon). However, all of them have at least one damaging move programmed at level 1 (say, if you use a cheating device to encounter a low leveled one in the wild, it will have a damaging move). Also, all evolution Pokémon (except the ones in the list) have at least one damaging move at level 1.

In generation II, some Pokémon evolved by stones don't have learn any moves at all by leveling up (for example Victreebel or Raichu). Nevertheless, all of them have damaging moves programmed at level 1, so any "wild" or generic battle Pokémon will still have a useful learnset.

From gen III onwards, breeding and the then-novel move reminders basically boosted the importance of the "level 1 moves", especially for high-leveled evolved and/or Legendary Pokémon, so the more reason to assume that the "no damaging moves learned by leveling up" are more and more rare. I hope that helped a bit, please correct or feel free to add extra info.

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Forgot the source: all from Bulbapedia learnset sites.
OMG thank you so much!