I think I have finally gotten the most difficult build, and the final build that can possibly beat the game: Vampire + Paladin.
The only other maybe-possible buildsI can think of is Rat-King + Rats + Darrr/Paramedics
Builds I think are impossible include:
Gold Dragon + Alchemist (the GD increases damage too slowly)
Phoenix + Hydra (probably not because the phoenix and hydra wouldn't be next to each other often enough to consistently increase hydra's damage)
Mechanical spiders seems impossible to get even close to the 40 damage/square you need to beat the final boss. Cockatrice eggs also seems to increase damage too slowly.
Other builds I've already beaten the game with include:
Hell Hounds
Ascetic
Tinkerer + Soldiers (Mecha)
Darrr + Beasts (esp. Ferret) + Diversity Demon
Grey Goo + Gold Goo
Tax Accountant + 3x Paramedics + something above
Any I've missed?