My favorite unit overall is a Praetorian. It's hard to argue against a cheap Macemen with no melee bonus at Iron Working.
On top of being completely and utterly dominant before Feudalism, they are not obsolete until Rifling!!
My favorite non-UU unit is a War Elephant. Need Ivory (means not always available) and can be cost-effectively countered by Spearmen in theory. In practice, you build a Barracks and a Stable and give your boys CI/Shock. Amazing thing is Construction also unlocks Cats - watch enemy empires crumble before your eyes. War Elephants are also usable until Rifling especially as anti-mounted against Knights/Cuirassiers which still retaining some offensive ability.

My favorite non-UU unit is a War Elephant. Need Ivory (means not always available) and can be cost-effectively countered by Spearmen in theory. In practice, you build a Barracks and a Stable and give your boys CI/Shock. Amazing thing is Construction also unlocks Cats - watch enemy empires crumble before your eyes. War Elephants are also usable until Rifling especially as anti-mounted against Knights/Cuirassiers which still retaining some offensive ability.