I suggest finding the games first, playing through them slowly, and only then to go looking for annotations...
What are the best annotated GM games for learning?

Andre, that would be quite difficult and time consuming to do it that way.
There are thousands of very good GM games.

Andy Soltis' book The 100 Best Games of the 20th Century, Ranked might be one place to start looking. However, I don't know what the annotations are like.
http://www.amazon.com/Best-Chess-Games-Century-Ranked/dp/0786427418

I second the book by Bronstein about the Zurich tournament in 1953.
http://books.google.com/books/about/Zurich_International_Chess_Tournament_19.html?id=VUrggX89EDcC
Also one by Lasker, the tournament book St. Petersburg 1909.
http://chess.about.com/od/chessbooksandequipment/gr/stp1909review.htm
Andre, that would be quite difficult and time consuming to do it that way.
There are thousands of very good GM games.
My point is to get the INSTRUCTIVE games and play through those, before finding annotations. Classic games are best because they concentrate on a limited number of themes which makes it easier and more memorable to learn from.
I don't know what your OTB rating is, but Zurich 1953 helped me a lot when I was 1000-1300. Personally, Alekhine's book of his games wasn't so helpful for me, but I haven't looked at it since the turn of the millennium...I recently ordered Kotov's book on Alekhine's games, which a lot of Soviet-trained GMs recommend very highly, and I should receive it in a few days.
Other posters have mentioned other books, but I can't speak for those. Good luck.

Andre, that would be quite difficult and time consuming to do it that way.
There are thousands of very good GM games.
My point is to get the INSTRUCTIVE games and play through those, before finding annotations. Classic games are best because they concentrate on a limited number of themes which makes it easier and more memorable to learn from.
I don't know what your OTB rating is, but Zurich 1953 helped me a lot when I was 1000-1300. Personally, Alekhine's book of his games wasn't so helpful for me, but I haven't looked at it since the turn of the millennium...I recently ordered Kotov's book on Alekhine's games, which a lot of Soviet-trained GMs recommend very highly, and I should receive it in a few days.
Other posters have mentioned other books, but I can't speak for those. Good luck.
Are the Kotov annotations of Alekhine's games that are recommended by the Soviet-trained GMs algebraic notation? Just curious...

I am just saying it would be hard to find the instructive games and then to annotate them. I am sure there are games out there which are very good instructive games which are also annotated. A few of Fischers games for example.
I "know" there are games out there which are very good games and already annotated. Maybe they are on youtube?

Andre thanks! Now we have a list started which I will add these possible games
Deep Blue vs Kasparov 1966 game 1 [won by Deep Blue]
Botvinnik vs Capablanca World Championship game 1938
13 year old Bobby Fischer vs Donald Byrne [game of the century]
Kasparov vs Topalov 1999 [Kasparov Immortal Game]
[after the list is large enough we can choose the 20 best]

Scottrf, It was probably Kasparov's best game and Kasparov was one of the two best players of all time--so you can learn a lot.
But why avoid playing anyone?
Will also add Bernstein vs Mieses
and Capablanca vs Corzo 9th match game 1905
[I am terrible at making long lists due to a handicap--can anyone make a total list from what we have already?]

There are some excellent annotated game collections available.
Some of my favorites include:
Learn from the Legends by Mihail Marin.
On My Great Predecessors (5 volumes) by Garry Kasparov
Fire on Boardby Alexei Shirov
Taimanov's Selected Gamesby Mark Taimanov
Life and Games of Mikhail Tal by Mikhail Tal
V Poickakh Garmonii by Vasily Smyslov
Zurich International Chess Tournament, 1953 by David Bronstein
My 60 Memorable Games by Bobby Fischer
Art of Chess Analysis by Jan Timman

Oh, there's a really inexpensive, really great book that everyone should have:
The Mammoth Book of the World's Greatest Chess Games (1998) by Graham Burgess, John Nunn and John Emms. The first edition contains the 100 greatest games as voted on by the three authors, and excellent analysis of each game. A second edition came out a few years later and included 12 more games played from 1998-2004.
This is a must-buy book. When it first came out, it was only $10.95! The only problem with the book is that it will eventually fall apart from overuse!

There are some excellent annotated game collections available.
Some of my favorites include:
Learn from the Legends by Mihail Marin.
On My Great Predecessors (5 volumes) by Garry Kasparov
Fire on Boardby Alexei Shirov
Taimanov's Selected Gamesby Mark Taimanov
Life and Games of Mikhail Tal by Mikhail Tal
V Poickakh Garmonii by Vasily Smyslov
Zurich International Chess Tournament, 1953 by David Bronstein
My 60 Memorable Games by Bobby Fischer
Art of Chess Analysis by Jan Timman
I'd thumbs up Instructive Modern Chess Masterpieces by Stohl.
Thumbs down Timman's and Bronstein's. They're not bad books by any means, but IMO they don't deserve to be called great either. Maybe personal preference.

By the way, on topic, I think this a great idea. Unfortunately I don't think I can contribute :( Hope this list gets made though, I'd go over those games :)

I am asking for individual games but you can pick a few of the best games in some of these very good game collections.

Here are 4 more possibilities
Kasparov vs Deep Blue 6th game Kasparov only needed a draw but went for a win
Lasker vs Capablanca 1914
Fischer vs Panno
Fischer vs Tal 1959

Andre thanks! Now we have a list started which I will add these possible games
Deep Blue vs Kasparov 1966 game 1 [won by Deep Blue]
I'd love to see how Deep Blue beat a 3 year old Garry Kasparov!
Botvinnik vs Capablanca World Championship game 1938
The AVRO tournament was not a World Championship event. But the game is often considered the greatest game played in the 20th Century.
13 year old Bobby Fischer vs Donald Byrne 1956 [game of the century]
Here's a link to the game: http://www.chessgames.com/perl/chessgame?gid=1008361
Kasparov vs Topalov 1999 [Kasparov Immortal Game]
Here's a link to Kasparov's analysis: http://chess-club-slon.ru/art_kasparovbg.php
[after the list is large enough we can choose the 20 best]
Those are good, potentially instructive games. Can anyone suggest where to find good annotations of them?