Hello, I'm new to the community, hope I'm not too late responding. I've kind of eye-balled the S4 mini for a while but wasn't too crazy about having to use a laptop brick for what (in my eyes) is "supposed to" be a full-featured-but-portable computer. I was sitting on a HDplex 400W combo for a different build that... kinda fell through, and I wasn't going to buy a Skyreach until I found a 1660ti from Zotac that fits brickless in the case.
https://www.newegg.com/zotac-geforce-gtx-1660-ti-zt-t16610f-10l/p/N82E16814500462 - Zotac 1660ti
https://www.newegg.com/zotac-geforce-gtx-1660-zt-t16600f-10l/p/N82E16814500465 - there is also a 1660 if you want to save $50...... but I doubt saving money is what we buy S4's for.
Both of these cards are better than the 1060-6gb which is great because THAT was a solid card.
If you think you really want ray tracing though, check these out:
http://www.inno3d.com/products_detail.php?refid=416 - A 2060 from Inno3d that seems well liked by Josh, but hard to come across in the USA.
https://www.newegg.com/p/1FT-000N-001C6 - Also a 2070 however..... I think it is legitimately too expensive considering a 2070 is usually about $500.....
https://www.newegg.com/zotac-geforce-rtx-2070-zt-t20700e-10p/p/N82E16814500456?Item=N82E16814500456 - Another card from Zotac, a 2070 that will fit, but only if you take off the backplate and fan shroud, then put in different fans.
The size restrictions for brickless seem to be about 113mm in width (4.45 inches). You CAN put them in even with the PCI power connector and don't have to mod the card in the slightest. The most "crazy" thing you have to do is take the cover off the AC-DC ATX converter so the PCI cable will fit. Just DO NOT stick your fingers in there, PSU caps can store a nasty charge for several days even without power from the wall.
If you're not comfortable with that, a 1650 succeeds the 1050ti and is decent for gaming. No modding or PCI power cable required.
All the information for build types can be found here, and the build sheets include links to more parts, as well as videos on how to build them as well. They are just long, sometimes an hour and a half.
https://nfc-systems.com/boms I am only trying to shorten things a bit, all credit goes to the original creator.
The most powerful card that you can fit presently is a 1080ti but it looks like you can only get them used, and they still cost an arm and a leg..... And you also cannot put it in a brickless build, so I won't be linking that in this particular thread. At that point though you will probably need to do some pretty aggressive undervolting of both CPU and GPU, because you want to keep your AC power efficiency up as high as possible.