Generally, I find that the AI civs that make an unusual amount of units immediately suffer greatly unless they get paired next to someone that doesn't make any. In that case, they often become a runaway civ. Monty making 20 jaguars and zero libraries comes to mind and that's probably why Atilla is mentioned by other people (rams and horse archers with zero libraries).
On the other hand, some civs that favor tech (Korea and Babylon immediately come to mind) will blast everyone out of the water if they are left alone. They will either turtle with a tall, powerful nation, or rampage with technologically advanced units.
Lastly, Babylon is the best civ to have next to you if you are Rome. Bowmen are obsolete now, ballistas counter Walls of Babylon, and you'll get a capital with an academy present (basically stealing Babylon's UA). It works for other classical-warfare civs like Greece for instance.