A lot of it will depend where you live since a few states, CO, NH, ME, MI, SC, FL only honer resident permits. I'm in OR which has terrible reciprocity (and will continue that way until state Senator Floyd Prozanski is out on his ass - let's make that happen - but I digress). I currently have OR, UT, and ME permits. I will be getting FL, NH, NV, and CT in a few months (a single course that I will be taking satisfies the training requirements for all 4 - UT and OR too but I already have those). My UT and FL will give me most states that have reciprocity except for CO, NH, ME, SC, NV. Nevada's reciprocity is pretty limited and you may very well need to get a Nevada NR permit if you want to carry there. For CO, NH, SC technically my Florida permit would give me those states but since I'm not a resident of FL, it doesn't help. I can get a NH non resident but I'm out of luck for CO and SC. They don't offer non resident permits (well SC does but I need to own land there). ME has very limited reciprocity but it's very easy to get a non resident permit. I will also be getting CT which does offer Non resident permits but doesn't recognize any other states.
When all is said and done I will have 7 permits. OR (resident), NV, UT, FL, ME, CT, NH. That will allow me to be licensed in all states expect CA, HI, CO, IL, MA, NY, NJ, MD, RI, SC, DC, WI (until later this year where they will almost certainly have reciprocity with one of the 7 permits). Depending on where you live you may get CO and SC and you may get NH, NV, and ME without explicitly needing to get those non resident permits. I think technically I could get MA, RI, and NJ non resident permits but they are either almost impossible to get or so cumbersome and expensive to get that they aren't worth it. Once i have all 7 permits I think my UT one will be no longer needed since all other permits will cover anything that UT would give me. I'll still keep it. I like to have more than one permit covering a state from a reciprocity perspective that way if the reciprocity agreements change I'm less likely to be caught out.