Dragon Warrior 1:
I actually leveled up to level 30. I note that, after a certain point, Metal Slimes go down as easily as regular Slimes. (You need to have around 130 or so Attack Power for this to happen.)
Final Fantasy 1 (PSX version):
On Easy Mode, I leveled up a Monk to level 99, then, with the help of magic (Haste + 26 uses of the Giant's Glove), did 50976 damage to a Frost Wolf. (Frost Wolves have 0 defense and don't run away, making them the perfect subjects to do this against.) I then worked out mathematically that this amount of damage, which is consistent (RNG doesn't affect it), was the maximum possible without cheats (assuming no glitches I don't know about).
Etrian Odyssey 2:
I managed to get my Medic, when attacking a hedgehog (weak against Blunt) with Caduceus, to hit so hard that damage overflow occurred and the damage was significantly lower than expected. (IIRC, about 26k when, with lower stats, I had done around 58k)
SaGa 1:
On one solo playthrough, I got the bosses to run out of attacks, at which point they just die. This works for all bosses except Byak-Ko (infinite use SPHERE) and the final boss (who seems to be a special case). The invincible Su-Zaku would run away instead, but the game still gave me the boss death animation in that case.
Castlevania: Dawn of Sorrow
With the help of a memory corruption glitch, I beat Soma while playing as Soma. I believe the screen turned white and the game froze when I won, as the game developers hadn't expected it to be possible.
Baldur's Gate 2:
I killed the tutorial guy by using a cheat to get Greater Deathblow. (This could also be done by exporting/importing a character from ToB who already knows that HLA.) Of course, this softlocks the game as you need to talk to him to continue. Note that the Mind Flayer Shapechange trick doesn't work here because he is immune to *all* weapons, regardless of enchantment.
Final Fantasy 5:
On one playthrough, as soon as I reached Omega, I killed it before continuing with the game. (Omega is a superboss who you are meant to avoid, and who is meant to be an extremely challenging boss.)
Also, I got damage overflow to occur here as well. I believe I combined 2-Handed, Spellblade Holy (against an Exdeath Soul), level 80, and 99 Strength. (Level had to be 80 to demonstrate this; at level 99, damage was so high that, even after the overflow, I still did 9999 damage.) I also computed that, if it weren't for the damage cap and the game's use of a 16-bit integer for damage amounts, it would be possible to do over 300k damage with Goblin Punch.