A great example of a free online adventure single player RPG game is Legends of Zork. It's based off of an old RPG game that ran on the Atari. The storyline has been converted and streamlined into an online text based game, which features hundreds of locations for you to explore, along with plenty of loot for you to collect. In true RPG fashion, you create your character and begin exploring, encountering monsters and fighting them to gain experience and loot. Once you level up, you gain a skill point, which you can then use to benefit your character in a variety of different ways.
Legends of Zork is completely free to play online adventure RPG and offers all players 25 new turns per day. This limits the amount of actions you can take in one day, but you can also let your turns build up for a longer play session. Just be careful, as if your turns accumulate past four days worth, you'll begin losing turns. This is the developers way to ensure people return to play the game.
Check this game out at http://legendsofzork.com