I've wondered about this, and your question prompted me to do some research, with the help of the "Create License Map" tool from handgunlaw.us.
The first obstacle is the states that don't honor Utah's permit and only honor resident permits from other states. For those, you have to get a non-resident permit from each state. They are: KA, SC and WV. Except that KS and WV don't issue non-resident permits, so you can't get them. So gotta get SC.
Another obstacle is the states that don't honor anyone else's permits: CA, CT, DC, HI, IA, MA, MD, ME, NE, NJ, NY, OR and RI. So, to carry in those states you have to have a permit from each. Except that CA, DC, HI, NE and NY don't issue non-resident permits. OR only issues them to residents of WA, CA, ID and NV, so you can't get that one. You can get CT, IA, MA, MD, ME, NJ and RI.
That just leaves us with the states that do honor someone's non-resident permit, but not Utah's resident permit. Unless I made a mistake looking at the maps, there are no states in this category.
So, for a Utah resident to maximize CC permit coverage, you need: UT, CT, IA, MA, MD, ME, NJ, RI and SC. With possession of these permits, you can carry in all but CA, DC, HI, IL, KS, NE, NY, OR, WI and WV. Nine states plus DC.
According to handgunlaw.us (where all of this information came from):
CT is very close to "shall issue". Easy and can be done by mail.
IA is easy, but you'll need proof of a training program including a firing range component.
MA is easy, but requires a course from an MA certified instructor and must be renewed annally.
MD is nearly impossible to get. You have to show cause, with supporting documentation (police reports).
ME is easy and can be obtained by mail.
NJ is very difficult. You have to show cause, and they'll still say no.
RI isn't difficult, but requires that you show cause.
SC isn't difficult, but requires owning real estate in SC.
So, throwing out the ones that require showing cause or buying real estate, you should probably get UT, CT, IA, MA, ME, which will allow you to carry in 37 states. You might be able to get RI and SC permits to make that 39.
As Dave pointed out, though, each of those additional 4-6 permits only gives you coverage of the issuing state, so it's really not worth the effort unless you know you're going to be in that state.