^No he's not done it yet. He's close though. At the moment his rating is 2875-ish. His peak rating was 2882 I believe. Kasparov's peak rating was 2851, and Fischers 2785.
But please bear in mind that Elo ratings are not an absolute measure of chess strength. They are a relative measure. So when you consider that when Fischer had his peak monthend rating (and note the distinction, because FIDE only ever published calendar monthend numbers, while we now have websites that show live ratings) of 2785 at a time when the next best rating - I'm guessing it was Spassky at the time - was some 120 rating points below him. At Kasparov's peak his rating superiority to the next best player peaked at something like 82 points. Carlsen's peak gap to the next best is about 74.
What we can say however is that in all likelihood Carlsen's sustained gap to the next best may well be the best in history already. I don't have the numbers to hand. What is even more impressive though is the fact that Carlsen is competing in an era where everyone has access to huge computer databases so it's very impressive that he is able to maintain a gap such as it is versus the field. Add that to the fact that chess is even more widely played at a very high level now than even in Kasparov's time so Carlsen is definitely dominating a deeper population set. By any measure what this young man is doing is practically unheard of and he hasn't even reached his peak strength yet.
What I find even more frightening is that, to date, he has not really focused on using home preparation to beat the opposition. This recent tournament might be the start of something new as he used home prep to beat Kramnik who used a Berlin defence against his e4. That probably doesn't make much sense to you, but trust me when I say that Kramnik has been the premier proponent of the Berlin defence and he used it to defeat Kasparov in the World Championship. Kramnik is typically very highly prepped while Magnus isn't, so for Carlsen to beat him like that is really scary.
Carlsen is a player who has simply wanted to avoid home preparation and get a playable position in a middle game with the confidence that his superior chess ability will win the day. He's a throw back in a way. His style is the complete opposite of what you would expect in an era that has computer resources available to them. In short this is one stupendously strong chess player, and I am already of the view that he belongs in the the rarified selection of all time great players. A list that I would limit to: Kasparov, Fischer, and possibly Lasker.