Here are a few tweets from an Akiba shop on the CPU cooler compatibility for the ASRock H110M-STX ASRock inside the DeskMini 110:
They say both Silverstone NT08-115X and Noctua NH-L9i have no clearance issue with on-board components (barely clear off the silver heatsink on the right side of the CPU socket) but NT08-115X is 48mm tall and is a bit too tall to fit inside the DeskMini 100 case, a 45mm tall cooler will probably fit. I threw them a question regarding the 47mm tall Cryorig C7 to see if they'll take time to check that out.
ETA: They checked C7 upon my request:
They said three issues shown in the photos: 1) simply too tall; 2) the C7 backplate not fitting with the stock aluminum backplate; 3) mounting screws too long.
Also, there's already someone else who tried an NH-L9i heatsink (23mm tall) with the stock 14mm thick fan replaced with a 25mm thick fan (with the total 48mm tall), and he reported that this combo actually did fit in the case once he trimmed parts of the fan frame that hit the hooks on the case when sliding the mobo tray back in. He said the hooks protrude by 2 to 3 mm, so 45mm is the max height of a cooler that can be tucked in the case without modding. (Those circled with orange in the image below.)
I personally don't like the way ASRock (or whoever the case was outsourced to) designed the case for opening/closing, as it looks like the top panel isn't removable by simple unscrewing, which means this tray sliding system prevents us from sticking our fingers in the case while sliding the whole system back in to keep the floating (unguided) portion of the front power switch cable away from the fan on the CPU heatsink. I try to believe they chose the length of the cable so that the floating portion will not reach fan blades, at least with the stock Intel cooler, but I don't completely trust them.