How often should you climb per week?

The amount of times you should climb per week really depends on you and how conditioned your body is to climbing. If you have been climbing for a few years, and you are generally healthy, you may be fine climbing 4 times per week or even more. If you are a new climber, climbing 3 times may be your limit before you start experiencing slight or major injuries, and you may even be able to climb only twice a week.

In the end, the amount of times you can climb per week really depends on you. If you come from a background in sports, and you are a new climber, then maybe you can even push it and go for 3-4 sessions per week, but this is considered risking it because climbing generally uses very specific muscles that most people, even those coming from other sports background, are not used to, and if you overuse these muscles and don't give them the adequate amount of rest, you could just be risking injuries.

The best way to find out is to just experiment. Try going twice a week while having a rest day or two in between, if you are ok with this, try adding another day in, and just keep experimenting with the amount of climbing days and the amount of rest days in between.