  In a shallow near-infrared survey of the dwarf Irregular galaxy, NGC 3109,
near the periphery of the Local Group, we have found eight Mira variables,
seven of which appear to be oxygen-rich (O-Miras). The periods range from about
430 days to almost 1500 days. Because of our relatively bright limiting
magnitude, only 45 of the more than 400 known carbon stars were measured, but
none was found to be a large amplitude variable. One of the Miras may be an
unrecognised C star. Five of the O-Miras are probably hot-bottom burning stars
considering that they are brighter than expected from the period--luminosity
relation of Miras and that, by comparison with theoretical evolutionary tracks,
they appear to have masses >~4 Msun. A census of very long period (P>1000 days)
Miras in the Galaxy and Magellanic Clouds is presented and discussed together
with the newly discovered long period, but relatively blue, variables in NGC
3109. New $JHKL$ photometry is presented for three O-rich long period Miras i n
the SMC (including a candidate super-AGB star).
