Currently
https://www.computerchess.org.uk/ccrl/4040/
Historically
There are arguments for various players being the greatest of all time.
IMO Kasparov.
If Carlsen stays #1 for ~15 years, and wins ~5 world championship matches, then I'll consider changing to Carlsen.
Many men, many minds.
Many men, many minds.
Many men, wish mate on me
Blow smoke in my eyes so I can't see
I'm tryin to be what I'm destine to be
But hustlers tryin to take my king away
Many men
Many, many many many men
Wish mate on me
exactly.
Bobby Fischer's rating for the period when he demolished the other top contenders in unheard of fashion 6-0, 6-0, 5-3-1 is the highest rating EVER even though modern ratings are inflated compared to those in the 1970's and then he toyed with Spassky even though Fischer's antics made him start down 0-2. He lost only one game in the candidates matches - and he was ill when that happened!
No one has ever dominated chess in this way.
In my opinion the bet in history is Kasparov,but best today I think is Aronian and Magnus.
I think Carlsen could beat any chess champion from history. So could Naka, Karjakin, Caruana, So, Aronian, MVL, etc. These players have a huge advantage over Kasparov or Fisher in that they learn from computers. If Kasparov and Fisher were 20 years old today, they would probably be the best in the world, having the same advantage. But 1990 Kasparov or 1970 Fisher were not as good as todays players.
I think Carlsen could beat any chess champion from history. So could Naka, Karjakin, Caruana, So, Aronian, MVL, etc. These players have a huge advantage over Kasparov or Fisher in that they learn from computers. If Kasparov and Fisher were 20 years old today, they would probably be the best in the world, having the same advantage. But 1990 Kasparov or 1970 Fisher were not as good as todays players.
Ok, but Anand and Kramnik have been hanging around the top 10 all these years, and Kasparov, in spite of being retired for so long and his age, did fine in that St Louis blitz vs the contemporary top players.
You want to say Carlsen's modern play > Fischer's historical play, maybe so, but we can't really say the same about Kasparov.
More than engine though, today's players have access to information and opponents. Someone in Norway could get coaching via skype from a Russian coach, for example. And you can have a million game database delivered to your door while you sit on the couch. Engines are useful for finding blunders but don't teach you how to play chess.
Ok, but Anand and Kramnik have been hanging around the top 10 all these years, and Kasparov, in spite of being retired for so long and his age, did fine in that St Louis blitz vs the contemporary top players.
I don't get your point. Kasparov, Kramnik and Anand are now learning from computers but also ageing. They were all great champions, and still great players today. So?
You want to say Carlsen's modern play > Fischer's historical play, maybe so, but we can't really say the same about Kasparov.
Why not? Garry was great but not greater than a chess engine. The players now know more than he ever did about best moves and taking advantage of inferior moves that only a computer can show you.
More than engine though, today's players have access to information and opponents. Someone in Norway could get coaching via skype from a Russian coach, for example. And you can have a million game database delivered to your door while you sit on the couch. Engines are useful for finding blunders but don't teach you how to play chess.
All those things you mention are really just more benefits of living in the computer age. Engines absolutely DO teach people how to play chess.
Ok, but Anand and Kramnik have been hanging around the top 10 all these years, and Kasparov, in spite of being retired for so long and his age, did fine in that St Louis blitz vs the contemporary top players.
I don't get your point. Kramnik and Anand are now learning from computers but also ageing. They were both great champions, and still great players today. So?
Kasparov isn't much older than them, he used engines too 
In fact when he did that breif coaching with Carlsen he was surprised how little Carlsen knew about how to use the computer.
You want to say Carlsen's modern play > Fischer's historical play, maybe so, but we can't really say the same about Kasparov.
Why not? Garry was great but not greater than a chess engine. The players now know more than he ever did about best moves and taking advantage of inferior moves that only a computer can show you.
Except he played them, and they didn't crush him. This in spite of his age and number of years being retired!
More than engine though, today's players have access to information and opponents. Someone in Norway could get coaching via skype from a Russian coach, for example. And you can have a million game database delivered to your door while you sit on the couch. Engines are useful for finding blunders but don't teach you how to play chess.
All those things you mention are really just more benefits of living in the computer age. Engines absolutely DO teach people how to play chess.
Well you said they learn from computers, so I'm admitting yes, computers are a big deal, but engines not as much as people generally assume.
Just look at top tournaments. No one just copies the engine, because everyone knows that would be silly. All the time they look for practical advantages.
Also if you compare historical engine matchup rates (like Capablanca, who played 100 years ago) to modern players, there's not much difference. In fact Capablanca is one of the best.
The point, my friend, is that even after being retired for so long, Kasparov still plays on a competitive level with today's chess players. You say Kasparov is weaker then a chess engine, my question to you is: Who isn't? Computer training allows players to know what opening moves are better or worse then others, and also help understand where they made inaccuracies AFTER the game. During the game only human logic can figure out the best moves, as we are not computers and today's players can't calculate nearly as far as engines can. Also a computer can't show you how to play, it can only show you what to play. This means that while a computer might give you an evaluation of a specific position and teach you a specific best move, it won't teach you deeper chess understanding, you'll still have to practice tactics yourself, and memorizing openings is just as important as it was in the past. Magnus Carlsen is the best chess player of today, but will he be able to dominate the chess world for 20 years? Not very likely.
Except he played them, and they didn't crush him. This in spite of his age and number of years being retired!
I think he tied for 8th out of 10 players in both blitz and rapid. Not bad but not great. You are extrapolating that result to say he surely would have been 1st if he was in his prime. I'm not so sure. But it is a good example and worth pondering. It is also different than classical chess. Would modern computer training be more of an advantage in a classical game or blitz? I think maybe it is more of an advantage in classical because in blitz instinct plays more of a role?
Oh, tied for 8th, I thought he did better.
He was retired for ~10 years though while all the other guys play multiple super tournaments a year. Plus his age. I still think he was really impressive 
Carlsen was asked in an interview who might give him a challenge historically. He said Fischer or Kasparov in their prime would be difficult. How much of that is paying homage and what were his detailed thoughts, I have no idea, but I do think there's sort of a limit to human chess, and that the very best through history would give each other trouble provided they were allowed to catch up on theory.
Oh, tied for 8th, I thought he did better.
He was retired for ~10 years though while all the other guys play multiple super tournaments a year. Plus his age. I still think he was really impressive
Carlsen was asked in an interview who might give him a challenge historically. He said Fischer or Kasparov in their prime would be difficult. How much of that is paying homage and what were his detailed thoughts, I have no idea, but I do think there's sort of a limit to human chess, and that the very best through history would give each other trouble provided they were allowed to catch up on theory.
👍
Just imagine if Fischer or Kasparov had the benefit of modern computer analysis. I appreciate the brilliance those guys found, all on their own.
some say kasparov, others karpov and others carlsen but who is really the best?