Best way to run 3.3V circuit for long time using 3 AAA batteries?
If you use a CMOS flip/flop, the 'button press' can SET the
flip/flop, which can enable a low-dropout regulator like
TPS706  .   The regulator turns on, your MCU
goes through power-on operations, does its thing, and
turns the power off by issuing a RESET to the flip/flop.
When in 'shutdown', that TPS706 takes under 150 nA.

This is usually needless, though: many MCU chips have a
variety of on-chip functions to go into sleep mode.
A trio of alkaline AAA cells has roughly 1200 mAh of capacity,
so a modest drain (like, 30 uA) might be supportable.
30 uA will drain the cells in a bit over 4 years,
and 150 nA will take longer (so long that battery shelf-life
dominates).