I've just installed the latest (4.1.0.3257) logmein pro plugin on my 10.8.3 Mac. When you try to connect to it, you must enter the logged-in user's username and password. I have long passwords and as you try to enter it, the passwords already entered will get SELECTED randomly.
So if your password is "counting sheep 12345" you might effectively enter
coun [system suddenly selects all]
ting sheep 12345 [you press enter]
or
counting sheep 1 [system suddenly selects all]
2345 [you press enter]
and of course you can't log in. You can see the ****** entry getting highlighted as you type when this happens. If you type a few letters and wait, you'll see it happen.
I have the same version of the plugin on my other Mac which is 10.7 with no problems.