I played LoL's carrot-on-a-stick for about a year, but I just got sick of it. If I want to grind, I at least want to know there'll be an end to it.
I'm not sure what the carrot or the stick is in LoL. You just get to play, and if you want to play specific champions then you either pay real money or grind the IP. You certainly don't need all the champions, or even the majority of them.
JT, I'd recommend LoL (League of Legends). It's a DotA clone that cleans up a lot of the obtuse mechanics and is overall quite accessable. The way its F2P model works is that every week there are 10 champions who are available to play that week. If you want to play another champion you either have to shell out money or use the in-game currency you earn from playing. They release a new champion roughly every 2 weeks, though all champions fit into predefined roles, such as tank, support, mage, etc. Champions have alternate skins which can only be purchased with money. In addition to your champion selection, you can customize your play through the in-game skill tree and rune pages. The skill tree is a series of minor buffs that are focused on increasing offense, defense, or utility. The rune page lets you pick very specific buffs (one per rune) so you can either enhance your strengths or shore up your weaknesses. Runes can only be purchased using the in-game currency (the devs have stated they don't want to sell power).