what is the biggest chainring I can put on my mountain bike?
Do you have a crank that's easy to replace, like Shimano Hollowtech 2? It's actually easier to exchange those than chainrings. I usually ride a road-compact (50,34), but for long tours around the alps I exchange them for a MTB one (44,34,24). I have a non-indexed front-shifter, so I don't need to change settings. Triple crankset axis seem to be some 3mm longer than double road cranksets, don't know about double MTB - double road difference.

Anyway, beware that you might need to change the derailleur distance to bottom bracket - 44 chainrings are quite a big larger than 36 ones. With that comes readjusting the cable.

However, do you really need a bigger ring to keep up? I don't know your exact wheel size, but with 2m circumference at 36/11, 90 rpm you do about 35km/h