Start with the easier solutions above, like long handles on sockets, ring spanners, and penetrating oil periodically for a couple days.
Here's your last resorts... For if the easier solutions don't work.
* Impact driver - the least destructive. Do be super careful that you're undoing the nut, not tightening it (BTDT)
If that doesn't work, take the bike to a local workshop and ask nicely if they can use an air rattle gun.
* Not a bike, but I've had to do this on my car more than once. You could use an automatic center punch, just a little off center.
If you're getting nowhere, move on to something more vandalistic.
* Nut splitter This will cleave the nut in twain, and hopefully not damage the axle. Nut needs replacing.
* Heat - try heating the nut up with a butane torch. This will bugger up the temper of metals, so don't play the flame on the dropouts or anything you want to keep. Paint will likely get damaged by this. A yellow-flame lighter won't have the power, you need the tip of the lighter blue flame. Possibly combined with the auto centerpunch to shock things loose.
* Grinder Cut a slot in the nut without touching the frame at all. If that doesn't make it loose, then use a cold chisel or a flat screwdriver or the nut splitter above to remove. And knowing dremel cutoff wheels, you may need more than one. Possibly try a 100mm or 125mm grinder, but they lack the finesse of a dremel-style tool.
How's the axle? If its toast anyway, stick the other side in a bench vice and rotate the whole bike around.
You can also use the grinder to cut through the axle and nut.