You didn't list your gaming preferences.
I would point to MMOs like Guild Wars 2 that let you respec at will. Also, you can level a character in a lot of different ways. You can mix in pvp or stay straight PvE. But even if you only do PvE there are so many maps and regions each with their own set of unique events, you need not repeat a single quest outside of the intro while leveling multiple characters.
If ARPGs are your thing, you can try Marvel Heroes Omega, which is now on consoles, though the PC version is quite a bit better. It's a good free to play model as it only requires you to pay for vanity items. With respect to what you're asking for, there are multiple ways to level in this game as well and it's generally pretty quick to hit max level, as the majority of the game focuses on gearing and post-leveling bonuses. Respecing is always free.
Neither of these really features a strong story, although GW2 really does try, with seasonal story updates that are free as along as you log in some time after they go live and before the next phase begins. If you want story and endless respec, that's fairly rare. Andromeda is the most recent example, but as with most story based RPGs, quests are on rails, so you're going to be repeating yourself a lot. Elderscrolls and Fallout are the most free in the latter regard.
If you just want to change the look of your character, most MMOs offer those options, but usually on a pay basis. Games like Fallout and Skyrim, you can do it any time you like on PC just by pulling up the race menu. Hit the tilde key then type "showracemenu."
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