Rappelz

I have wasted way too many hours of my life on Rappelz. This one's a Korean made MMORPG. The game is completely free. If you feel like getting rid of some cash, you can use it to buy equipment for your character. However, most players never do.
The graphics are quite good, as you can see. It also features a nice assortment of cleavage. (I assume this is to distract your enemies.)

You can capture (or buy) a pet to train and take with you. You can also purchase a mount (they're pretty damned cheap) in order to move around the world faster.

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Type: Free, closed source (can pay for extra items, but most people don't)
Genre: MMORPG
Platform: Windows
Download From: http://rappelz.gpotato.com/

Battle for Wesnoth

I figure a good place to start is to talk about Battle for Wesnoth.
Westnoth is a Turn Based Strategy game with RPG elements.
The graphics aren't spectacular (though they're sufficient), but the gameplay more than makes up for that. Your characters can level up as you go along. Capture towns to earn more gold.

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Type: Open Source (Completely Free)
Platforms: Windows, Linux, Max
Modes: Single Player Campaign, Online Play
Download From: http://www.wesnoth.org