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Actraiser is indeed the pinacle of weird mixed genre games.
Basically a city builder / god sim mixed with platforming levels

Unfortunately the second game dropped the city building and was a boring platformer.

I believe there are a number of puzzle game / RPG hybrids of various quality.

There are some RTS games which allow an FPS or action game style control of single units, but I'm not sure that counts as unusual.

As a kid, I was wishing for two mixes :

1) Driving (à la Test Drive) + Shooting (à la, dunno, Operation Wolf ?)

It became a standard feature, in GTA-likes or open shooters with vehicles like Borderlands.

2) Base building (à la Evil Genius) and infiltration (à la, dunno, Prince of Persia or Mark of the Ninja ?).

It doesn't quite exist. But I was in awe when UFO/X-Com battles took place in your own base, and reflected its layout. Some games such as Dungeon Keeper also allowed you to visit your own lair. And many multiplayer builder games allow you to build defenses and infiltrate each others castles, I suppose.

Apart from that, I'm still still waiting for an Old West fallout2-like RPG, but that's more a mix of styles/settings than gameplays.