If all went 'well', not the password is stored, but only a hash of it. Not knowing the hash method will make recovery very hard, especially if a generated password is used.
A rainbow table attack could work. Very fast and efficient. The only problem is downloading the rainbow tables, but that should just be a matter of searching.
Other than that (or the dump method), I suppose you could write a program or script to brute force the password if you can come up with a set of passwords or password patterns that you usually use.
If you use patterns, e.g. <somebody's name><4 digits><2 symbols><obligatory curse word>, and know that each of those has 10 possibilities that you usually use, then you can store them all in a database and do a cross join to get the full set to use in a brute force attack. I've done it before, and it's pretty effective. Or you can just write up a program to do the cross join from string arrays for you. That's just a bunch of nested loops, so it's probably about as fast to write a custom program as to actually do it from a database.
However, I don't know of any MySQL password recovery tools.