I know almost nothing about optics, but out of curiosity I am trying to educate myself.  It seems to me as if most telescopes are reflecting telescopes while rifle scopes seem to be exclusively refracting telescopes.  I can see why this is the case, but I wonder...  Has anyone gotten creative and tried to make a reflecting telescope optical arrangement into a rifle scope?  Would it even be possible?

For one, reflectors have inverted images. There are other reasons to keep it simple

You can put a prism in the rifle scope to reverse the image. The reflector may be a bit bulky and not handle recoil well but you could build it.

Reflector vs high index glass is no competition until you get to telescope sizes larger than can be handheld, or are fine with a fat, cheap, low quality optics.

They don’t make reflector telescopes because reflector optics are better, they make reflector telescopes because the lenses for the refractor telescope of the desired aperture size get to be prohibitively expensive, and heavy. The bad out of focus artifacts “bokeh” of a reflector is limited by the usual extreme range (near infinity) it is used at. The loss of some of the aperture for the front reflector just means a still wider optics (which would put the sight even higher above the barrel than an equivalent refracting sight).

Really advanced reflecting telescopes can also have a bed of linear servos to bend the main mirror to correct for atmospheric scattering.

They're not scopes exactly, but some clip-on night vision devices (PVS-24LR, PVS-27) use catadioptric systems, which include a mirror.