When a game takes on the Free to Play design they are forced to make a decision about how hard they will push the sales portion of the game. When this is handled poorly the game will frequently receive the moniker Pay to Win. In Pay to Win games a gamer can encounter some small portion of the game but in reality this is more like a Free Trial than an actual free to play game. Players are restricted to limited content and when trying to play with paying customers their power level is so substandard that they can't hope to compare. This drives anyone actually interested in the game to be required to pay to keep up.
Even worse it is also possible that games give such advantages to paying customers that they can continue to up their power level as they pay more. While I'm willing to throw some money at a game that I'm having a good time with, I'm not interested in trying to compete with a single computer professional with a gaming addiction and a $120K a year job. If someone wants to throw $1000 per month to put them on top of the game, they win, I lose. I don't even want to play that game if that's who I'm forced to play against.
While it seems much like this medium is doomed, there are a number of companies actually doing it right. A friend of mine tells me these are referred to in the industry as "Not Evil Companies". Perhaps the most successful of these is a company called Riot Games that makes the surprisingly addictive League of Legends. Yes it's free, and yes it's worth every penny you don't spend.
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If you were paying attention, you may have realized that above I said that you purchase champions which is true to a certain extent. There is a rotation of 10 free champions and new champions appear to be released on about three to four week basis. You are able to pay money to purchase champions, or you can play the free ones. Also, by playing games you earn points you can also use to purchase additional champions without paying any fee. Playing a couple games (20-60 min usually) a day one could reasonably afford each new champion as they came out. Older champions are generally discounted while newer ones tend to be more expensive. Considering you only play 1 champion each game having a vast selection of 50+ champions will probably dilute your skill compared to focusing on a couple of champions you're particularly fond of.
In this model, everything you need to play the game can be obtained for free. It's not like there are champions you can only get by paying money which are prone to owning all the free to play champions. Nor are there special potions which help you do better in game. While new champions are frequently purchased by paying real money, someone content to carefully pick and choose their champions and make use of the free to play scenarios could easily find great success without paying for anything, ever.
The only thing that you can only pay cash for are champion skins which alter the appearance of characters. While I'm occasionally jealous of specific skins that are particularly interesting or humorous, not having them doesn't make me worse at the game than another player. Perhaps I'm just not as pretty. Which is something I'm entirely familiar with in the world of getting things for free. If you're not paying for it, it may not look as nice.
While this is a single example of free to play there are also tons of fantastic games out there that are browser based that are entirely free to play. Several of these deal with complex topics such as physics and design. A number of my students started playing Captain Forever after I witnessed them playing a moronic block dodging game. Some of them played it for hours over the weekend without realizing that they were beginning to think about balance, symmetry, opposing forces, and torque. They just think it's a game.