In order to obtain something with reasonable processing power, MiniITX is the only option (if you come across x86 setup which is smaller - probably it will be a dedicated device).
I highly recommend using low-voltage CPU's from recent Core generation
(and their derivatives), such as Ivy-Bridge or Haswell Pentiums/i3's with TDP of 35W.
I use one (i3-3220T-35W Ivy Bridge) as a Home Theatre PC. I did try Haswell (bear in mind that it's different socket - LGA1150, instead of LGA1155 - for older Core gens) but it ran significantly hotter
with hardly any gain in the processing power, so I sold it.
You can get these Ivy low power processors quite cheap (used ones), motherboards too.
My setup looks like this (bad cable management, but this case is just too small - yet it holds everything - slim DVD-RW and even a 3,5 inch HDDs):
The CPU cooler is very slim (it needs to be in order to fit into this particular case), but luckily it stays quiet (which was the issue in Haswell setup):