Maxthon 2.5.6.349 (Beta) runs LogMeIn in full-screen mode, though there are still some quirks and unresolved issues.
I'm unable to interact with a remote PC in LogMeIn's standard "in-window" mode. The remote desktop appears in grayscale, and I can't interact with it at all (no clicks, no keyboard captures). When I switch to Full Screen Mode, the remote desktop display switches to color and expands to fill the available tab. As soon as I click on the remote computer's desktop (or interact with it in any way), I enter LMI's full-screen mode, where I can work with no problems. On exiting full-screen mode, the remote desktop switches back to black-and-white and can no longer be clicked into.
Also interesting: If I detach the tab where LogMeIn is running, I can't interact at all. (I can't even switch to full-screen mode; all I get is a momentary flicker, and then I'm kicked back to in-window mode with the black-and-white remote desktop displayed.)
For what it's worth, this is workable enough for me to make 2.5.6.349 my default browser.

Hopefully future releases will be even more compatible with LogMeIn and other remote computing service sites ... !