What multi/many-core (micro)processors/controllers should I use for "embarrassingly parallel" computations?
Build a new machine

For now the most affordable solution for this is probably a PC with a good GPU. Try this configuration which is reasonably powerful, have no bottleneck and reasonably affordable (small form factor machine):


Intel Xeon E3-1231v3 processor (3.4GHz quad-core with hyperthreading)
Gigabyte GA-B85N-Phoenix mini-ITX motherboard
Kingston KVR DDR3-1600 RAM 8GBx2
AMD Radeon R9 290X GPU (or if you can afford it, R9 295X2)
Kingston SSDNow V300 120GB SATA-6Gbps SSD


Or if you have a good gaming rig, use it. The machine listed above is actually a suggested SFF gaming rig.

Another rig, a hell lot more expensive, a hell lot larger, more than three times as fast as the previous rig, and still no bottleneck at all:


Two Intel Xeon E5-2643v3 processor (3.4GHz hexa-core with hyperthreading)
Asus Z10PE-D16 WS dual-socket motherboard
Kingston KVR DDR4-2133 ECC RAM 16GBx8 (or if you can afford it, 16GBx16)
Two AMD Radeon R9 295X2 cards in CrossFire configuration (4 GPUs total, and if you can manage it, get three of them)
Intel SSD 730 Series (PCIe NVMe card)


Repurpose old systems, lots of them

Also if you can hunt down lots of old PCs for the cheap (e.g, my school is selling their 100+ Core 2 Duo PCs at $15 each after them being replaced by new i5 machines) you can whip up a few tens of them and throw them into a Beowulf cluster, if you know how to put those clusters into work.

For example if you whipped up 30 of those Core 2 Duo systems with 512MB RAM each, you end up with a cluster with 60 cores and 30GB of RAM.

The single-machine supercomputer

If you can score a Sony PlayStation 3 that can still run Linux, that beast will be able to help you a lot. By the way, according to USAF, put 1790 of them together in a Beowulf cluster and you get a TOP500 listed supercomputer.