There is no definitive number of times you can resole your climbing shoes. It just depends on the condition of your shoes, and on the fit you want vs the fit of the shoe right now.
Based on the different online climbing communities, the average number of times climbers get their shoes resoled is 4, though, some even do it 6 times.
In the end, it depends on the condition of your shoes. If the shoes are in relatively good condition, but just have holes in the toe area, then it's more than likely a good candidate for a resole, even if it is the 5th or 6th time. If the shoes are just very used up and beaten to the point where you have holes in other places besides the toe area, or the shoe is torn or just in bad condition, it may be time for a new pair of climbing shoes.
In addition to the physical condition of the shoe in terms of damage, it also depends on the fit you want to have. Climbing shoes made with leather tend to stretch quite a bit. The longer you have them and use them, the more likely they are to stretch. After a few years of use, many climbers feel that their shoes become "baggy" and not as tight as they were. If this is something that you actually want, then it's ok, but, if you prefer a tighter fit, and the shoe is already getting to be too loose, then it may be time for new shoes.
A resole does tend to make climbing shoes feel tighter sometimes, but this is not a consistent occurrence, and the tight feeling may only be in the toes, while the rest of the shoe still is quite loose. The tightness also usually only lasts for a short period of time.
All of my climbing shoes are synthetic, and I have resoled them all twice so far. If they will be in the same condition as they are now, just with holes again, then I will continue to send them in to be resoled until they are generally too used and damaged.
There isn't really a set limit for the amount of times you can resole them, it just depends on the condition of the whole shoe. If holes in the front of the shoes are not the only issue with the shoes, then will a resole help? If the additional issues can be fixed along with the resole, then it may be worth resoling.
If there are no additional issues with the shoes, and you just have holes in the front, and the shoes still fit well and are tight enough, and of course you want to keep wearing them, then there isn't really a defined limit to the amount of times you can send them.
If you send shoes to be resoled, and they cannot be, then you will probably get a call telling you that they can't be resoled anymore.
