OWC Express 4M2
The OWC Express 4M2 as the name suggests allow connecting four M.2 key M SSDs via Thunderbolt 3 and is shockingly cheap at 320 USD, the Netstor NA611TB3 is 360 USD and it only gives you two. Or open box for only 280 USD.
Since the PCIe switch is this cheap, you could do something crazy if you need a lot of parallel computing power (and are not constrained by host-to-card bandwidth): you could mount it in a Nanoxia Hydra II (because that one allows you very easy access to the PCIe slots) with two R43SR and two R43SL risers for four GPUs. Three from https://www.ebay.com/itm/282939143012 and one from http://www.era-adapter.com/m2-keym-to-pcie-x16-extender-cable-rightward-r4s-r3r-p-303.html because I am not sure 10cm is enough for all four. Chassis is 140 USD, four cables are 135 USD, a EVGA SuperNOVA 1300 G2 80+ GOLD PSU is 130 USD, 4M2 is 320, altogether 725 USD even with a healthy amount of fans it's less than 800 USD. The 3slot solutions which are physical three slots are 1000 USD, anything similar from cubix or netstor would be at least three times as much.
Lenovo ThinkPad 25 -- GALAX SNPR TB3 1060 -- Lenovo Graphics Dock -- Benq BL2411PT - - two PackedPixels - Dasung not-eReader backer