Regarding the buffering, keep in mind that a double-buffer scheme is a necessity in order not to generate audio glitches. Since a typical looper use will be playback + recording at the same time, that's at least 4 times a single audio buffer that's required here. If you use SDRAM for your main storage, only relatively small buffers will be necessary - typically a couple ms worth of audio should be more than enough. If you're using Flash/SD cards, you'll need much larger buffers - you may find that you don't have enough RAM on a given MCU to do this.
Just a couple figures: at 16-bit, 44.1 kHz: 1 ms of audio => ~88 bytes. So any SDRAM-based solution can be implemented even on an MCU with pretty limited internal SRAM. If you're using an MCU that can directly stream data from SDRAM to an I2S peripheral, you may even not need any intermediate SRAM buffer.
100 ms => 8820 bytes. Times 4 => 35280 bytes. Consider MCUs with at least 48 or 64 KBytes of SRAM. I think 100 ms of buffering would be a good starting point if you're using SD cards - of course using a good, class 10 card should get you lower max latencies so you may be able to use smaller buffers.
Regarding SD cards, as I said before, even though they may look like a simple and cheap (at least for an amateur design) solution, you may encounter more problems than you think trying to use them. You'll need proper "drivers" and at least a minimal filesystem. Thing is, IME, most available and free ready-made libraries for that are crap one way or another: very lousy performance, bugs, sometimes hard to figure out... In the same vein, many free filesystem libraries you'll find will also have poor performance. Writing that all from scratch is of course possible, but is definitely not trivial. SO getting it right and robust may not be as easy as you think.
Anyway IMHO, as I suggested earlier, I think starting with a small dev board with some Cortex M4 or M7 MCU, SDRAM and audio codec should be relatively straightforward. You could also consider an FPGA-based solution as some have suggested, but unless you're experienced with FPGAs, I wouldn't recommend it. Besides, a looper will required some UI to be useful, and implementing a decent UI is always much easier on a MCU IMO.
As to what existing pedals typically use - I have never torn one down, but I'd be pretty sure most of them use some kind of MCU or DSP, along with SDRAM. Depending on the features and the manufacturer, I wouldn't be surprised also if some pedals from the big players, such as the Boss one I mentioned, used some ASIC instead of an off-the-shelf MCU/DSP to further cut costs down.