ColoradoGirl, one published electrolysis study found armpis took about 10 sessions and 27 weeks on average.
Hairfacts: Urushibata (1995) I'm not aware of a laser study on armpits that kept as meticulous data, but many women have found laser to be fairly quick and easy.
Doing your own armpits is a little hard, especially with your non-dominant hand.
Barrester's comment about going as you have money is good if you can keep a fairly regular schedule. To increase effectiveness, you can wax in between. That way they are always treating actively-growing hairs. However, the best thing to do is shave between treatments and go regularly. Practitoner skill is vital to making sure you're done as quickly as possible.