My GPA is a 3.849… which could be rounded to 3.85…which could be rounded to 3.9…is that allowed?
It’s a bit of a stretch but 3.9 seems much better than 3.849 which I can make any decimal place as I am putting this on my own handmade resume.
Is it too much? 3.849, 3.85, or 3.9?
No, it can’t, it’s rounded to a 3.85. At some colleges, they may truncate instead of round, in which case it’s a 3.84. However, in no instance is it a 3.9.
^Right. Also, your school’s transcript might display your cumulative GPA rounded to some number of sig figs (usually 2, e.g. 3.8); if so, use that.
Why not round it to a 4? That makes as much sense as 3.9. 
Joking, of course. You have what you have. Report it as it is reported on your transcript.