A few weeks ago I bought a new (albeit cheap) self assembly bike from a local store, this bike to be exact http://tinyurl.com/gw77p9t . After about two weeks use (one and a half hours of cycling on pavement five days a week) the left crank started to creak, came loose and eventually fell off. I cleaned the crank and applied new grease which stopped the creaking and prevented it from falling off but the crank still came loose so as to make cycling uncomfortable. I then bought a new crank arm (this crank arm http://tinyurl.com/zz3mjzr ), took off the old crank, cleaned and greased again only to find the same problem. I can't imagine what else the problem could be, any suggestions?
A loose crank arm can damage both the bottom bracket and the crank arms. In your case, since you've run the crank arm til it fell off the bike, it is likely you ruined the bottom bracket. If your crank arm is loose, immediately stop and tighten it.
You will need to likely get a new bottom bracket and possibly yet another new crank arm. Replacing a bottom bracket requires a special tool (bottom bracket tool) and a compatible bottom bracket (which will typically be marked on the bottom bracket itself). You may want to leave this job to a bike shop.