LÖVE

Now that there is borderless window mode, how feasible would it be to let the draw surface itself be transparent?

As in, when you set background color to 0 alpha, you see your desktop.

Perhaps even moving the window itself from within LOVE. Desktop toys made in LOVE? The applications would be hilarious.

Unfortunately that's not possible right now. I believe it could be implemented in OS X (I haven't actually tried though), but I don't know about Windows and Linux. It would also likely have a relatively large performance impact.

Can be done with garden variety Lua BUT I've found that background alpha effects foreground alpha.