[Grejak]

²⁴⁴Pu is the most stable plutonium isotope with a half-life of 8.08 x 10⁷ years.  Since the isotope is an even/even nucleus, the ground state spin and parity is 0+.  It will either decay via spontaneous fission or alpha decay to a 0+ or 2+ state in the daughter.  None of the decay modes is strongly forbidden.  The 0+ to 0+ decay is allowed, in fact.  Furthermore, if you accept the shell model at face value, then the last proton level for ²⁴⁴Pu would be a partially filled 2f_7/2 with 2 protons whereas ²⁴⁰U would have a filled 1h_9/2 highest level.  For neutrons, ²⁴⁴Pu would have a partially filled 1i_11/2 while in ²⁴⁰U the same level would be half-filled.  I assume that a nucleus with a filled level is more stable then one with a partially filled level, although I am not entirely sure about how partially filled levels compare to half-filled ones.

Most of that points to a quick decaying ²⁴⁴Pu, so why is it so stable?

One answer could be that the high N/Z ratio for the nucleus decreases the probability of finding a formed alpha particle in the nucleus.  The decreased charge density at the surface compared to other Pu isotopes will also lead to a lower alpha energy of the emitted alpha.  Both of these factors should result in an increase in the half-life.

Likely it is the combination of many factors.  Any ideas what else to include?