Team Fortress 2 is probably the gold standard for F2P shooters.
The MMO market is full of F2P games of varying quality. The Old Republic is worth it for the story. I'm fond of The Lord of the Rings Online, but I have invested money into it. (though you can actually earn their version of premium currency via gameplay)
More low tech. but damn near any RogueLike if those are to your liking. (i'm a fan of A.D.O.M. going back over a decade, DoomRL is also fun for quick bursts)
Never played Warframe but it seems popular.
Planetside 2 is making strides to not try and murder even high end systems and is actually playable for most people now.
Leader of the Elder Cult of Jiub. Wow. Kenny's a Japanese Princess..