Is it possible to get rabies if a dog licks your arm with no cut?

Theoretically yes: It is theoretically possible to transmit rabies from infected saliva of a rabid animal to your mucus membranes without a cut. However this is very uncommon. If u know dog, find out if it is up to date with rabies vaccine or have dog evaluated. Sure beats getting multiple rabies vaccines.

Extremely unlikely: It is extremely unlikely if there was not an open wound. You should wash the arm throughly just to be sure. http://www.cdc.gov/rabies/exposure/type.html