Civilization: Beyond Earth does have a demo, but it's limited to 100 turns - not many to feel how a Civ game ticks. Given lukewarm responses to it, you might want to see more. Here's your chance: the full game is free for everyone to play on Steam until Sunday evening. Saints Row IV is also running a free trial weekend but surely you already know whether you'd like to be a superpowered scamp flying around cyberspace and punching aliens. Both are on sale too.
Clicking this here will get Steam cranking away on downloading Beyond Earth, while this one over here will get Saints Row IV going. You can play 'em up until 9pm (1pm Steam time) on Sunday, at which point they'll become inert until another free weekend rolls around or you buy 'em. That option is a whole lot friendlier this weekend.
SR4 is £4.24 on Steam until Monday (longer than the free weekend runs), or it's £6.24 with oodles of DLC, or £9.99 for Saints Row 2, 3, and 4, along with so much DLC. Or if you want something marginally more down-to-Earth, SR3 and its DLC is £4.24; I can relate to arguments for liking 3 more than 4, but I like the superjump so much.
Demos are great and I do miss them being commonplace, but SR4 and Civ are games far better shown off in slices bigger than publishers want to offer permanently. I'm glad they at least have occasional weekend trials rather than no demo at all.