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Its possible, there are some videos of homemade ones on the youtubes. I haven't found a site/forum dedicated to guitar effects with arduino that would be a good starting place though. Just some partially complete instructibles or youtube demos. Arduino seems like it would be a great tool to get into making some digital effects like a true bitcrusher, loopers, etc.
It's possible with AVRs (which is the same chips Arduino uses), but sound quality is limited. I can't speak for the Arduino clones that use more powerful ARM microcontrollers, but the regular 8bit ones will limit you at around 8bit 10khz sound or so. Maybe slightly more if you have serious assembly ninja skills. I built one a few years ago.
The WTPA sampler also uses that same lofi hardware, though I think it's a lot of assembly, and not Arduino's language.
A lot of the work is ADC prefiltering and DAC/PWM postfiltering as well.
I moved up to PIC32 micros for hi-fi looping/sampling effects. PIC32 and ARM micros will easily do 24bit 96khz sound, or even multichannel 16bit 44khz. But with these faster micros comes whole new hardware challenges.
I'd post more, but I can only use quick reply here at work. I wrote a guide for another forum, maybe I'll make a page or something from it.