Square Enix is offering a free trial for Final Fantasy XIV: A Realm Reborn, giving interested players access to the first 20 levels of the fantasy MMO.
Other than that there are a few restrictions on what you can do in game; you can't pay for optional extra services like additional Retainers, and you'll be unable to log in to the Lodestone and the official forums.
There's a few other restrictions on the amount of gold you can earn and some other bits and bobs, but nothing too debilitating considering it's only a short trial. There's a more in-depth FAQ for the free trial over on the game's official website.
Final Fantasy XIV is set to receive its first expansion, Heavensward, in Spring of next year. Grab the trial client here.