How many times can I resole my shoe?
Three, maybe four.
Usually the stitching will blow out on a climbing shoe if you've worn it through several re-soles. That can also be repaired, but it's a problem of diminishing returns.
Another terminal failure might be holes in the top of the shoe, from dragging your toes or jamming rough cracks.
To best extend the life of your shoes, take good care of them. Clean them off regularly with a damp rag (especially after climbing outdoors), and don't let them fester in the bottom of your gym bag.