Apology

[spinozza]

What is the most polite way to apologize to someone?

and by this, I mean "extremely polite"

[Bueller_007]

What is the most polite way to apologize to someone?

and by this, I mean "extremely polite"

Probably "moushiwake gozaimasen".
申し訳ございません。

There are also some much more complicated phrases, I'm sure, but that's a good one to know.

[Bucko]

Yeah I was taught that 申し訳ございません is the most polite apology. Roughly translates to "there is no excuse for my behaviour".