I think chalk is Calcium. If so, it's good for the body. But, if you slide you finger over the Chalk board, you may have a problem.
Pure chalk will not cause any serious problems on it's own, however prolonged ingestion can cause digestive issues as it will neutralize stomach acids. BUT the big issue is one of mental health. As this isn't a normal thing to do eating chalk this long is an abnormal behavior and a mental healt professional does need to be consulted before other abberant behaviors start to occur
Well I can't see the sources of chalk being very pure, you might get some sort of toxic buildup. Could also end up with a chlorine deficiency if that's even possible, since your body would need to make more HCl to neutralize the CaCO3. It could also cause respiratory problems such as reduced lung capacity and eventually lung cancer if you're inhaling much chalk dust as you eat it.