There isn't any limit to how many cycles you can do, though most couples will eventually be limited by budgeting constraints since it's quite expensive. A RE (reproductive endocrinologist) may also set limits based on their own professional assessment of whether or not the treatment will work. There are some unscrupulous doctors out there, but MOST REs want their patients to succeed, and won't let a patient waste their time or money on treatments that are unlikely to ever work.
Now, by "cycles," what they mean the entire process of IVF from stim meds, egg retrieval, fertilization of mature eggs, and then (hopeful) transfer of viable embryos. Protocols can vary, though. If you have a "fresh" transfer, you'll stim (for 8-12 days), have your retrieval, and then your embryo transfer all in the same month. If you have a "frozen" transfer, then you'll stim one cycle, give at least one cycle for your body to rest, and then the embryo(s) is later thawed and transferred at a another time.
I personally went through 3 cycles of IVF. I needed two ERs to get enough embryos to be screened via PGS (preimplantation genetic screening, which is an optional process that we decided to do), but that only led to 1 viable embryo that didn't work out. My third ER went much better, and after PGS we had 3 viable embryos. The first of those two also didn't work, and at that point I switched clinics to try a different method of frozen embryo transfer (FET). "Medicated" FET involves suppressing your natural cycle and doing everything with HRT, and I tried that twice without success. My third attempt was "natural cycle" FET, where my body was allowed to build an endometrial lining and then ovulate (with a little help) so that I was making my own progesterone. That was the cycle that worked. Clinically there is very little difference in the success rates between "medicated" vs "natural cycle" FET, but natural cycle just worked better for me. Overall, FET has a higher success rate than fresh transfer, and some clinics don't even do fresh transfers at all because of that.