I have a bog standard Deore 42/32/24 chainset on my bike, but as all the trails locally are intersected with road sections, I’d like to make the most of the tarmac bits and increase the gearing.

What size big ring can I go up to without effecting front mech function?

Ta

*swoons*

Real men have BIG rings !

44T defo
46T possibly – but you’ll also need to check that your rear mech has the capacity for the extra chain length.

Have you checked the small cog on your cassette? if its 12T, you could fit an 11T

Possibly 48 as I run Deore 48,36,26 on my MTB commuter but you should check the Shimnao tech docs to see if your front mech is compatible.
http://techdocs.shimano.com/techdocs/index.jsp
Shimano front mechs need to drop at least 12 teeth from big to middle to function correctly too I think.It’s all in the tech docs.