"He's unarguably the best" or "He's arguably the best"
Arguably the best means there are reasons to believe he is the best.
Unarguably the best means nobody can disagree he is the best.
Both are grammatical, but the second is a stronger statement than the first.
In he's arguably the best, the word arguably is being used as a hedging device. This means that it softens the strength of the claim and can help the speaker avoid being contradicted. The fact that you have said arguably actually prevents people from arguing with you about it! (You have already conceded that people may disagree about whether he is the best or not). It is functioning like a concessive device.
I don't know whether there is a formal terms for this, but the use of unarguably is exactly the opposite of a hedge. It is telling the listener that you are absolutely certain that he is the best and that you are so confident about it that you claim that the proposition is unassailable.
Both of these terms are used to modulate the listener's understanding about your confidence in the assertion. One limits your apparent confidence in your assertion the other increases it.
Both of your examples could be used with a different meaning, but "arguably" is more braodly used than "unarguably".
Each word of the English language has its specific purpose or meaning when people choose to use it.
Arguably is defined as a "sentence adverb" which means:
It may be argued (used to qualify the statement of an opinion or belief): ‘she is arguably the greatest woman tennis player of all time’
People argue because each has a different point of view and way of judging something or someone. If you use "arguably" in a sentence, you are admitting that "what you are stating" can be a topic of argument as the above example explains. Some people may think Martina Navratilova is the best female tennis player "ever", others may think Stefanie Graf is the one. It is arguable and it will "arguably" take years to reach a conclusion. There might be "no" end of this argument at all because it is your preference and how you view their records.
You use "arguably" when:
You know there could arise an argument because you know your statement is controversial and people might have a differnt view/idea about it.
You want to avoid any argument about this topic.
When you Google search "arguably best", "unarguably best", you get 182,000 hits for the former and only 6,890 hits for the latter. If you Google "unarguably worst", you get merely 274 hits, but 30,400 hits for "arguably worst".
You can't depend on the number of hits, but you can see "unarguably" is far less used than "arguably".
Adverbs like "absolutely", "certainly", and "surely" are far more frequently used than "unarguably". The reason why is arguable but you "arguably" don't want to use an "unpopular" adverb among many (which can mean the same) when you make a statement.