  We report scanning tunnelling microscopy and spectroscopy (STM/STS) studies
on one and two unit cell (UC) high FeSe thin films grown on Bi$_2$Se$_3$(0001).
In our thin films, we find the tetragonal phase of FeSe and dumb-bell shaped
defects oriented along Se-Se bond directions. In addition, we observe striped
moir\'e patterns with a periodicity of ($7.3\pm 0.1$) nm generated by the
mismatch between the FeSe lattice and the Bi$_2$Se$_3$ lattice. We could not
find any signature of a superconducting gap in the tunneling spectra measured
on the surface of one and two UC thick islands of FeSe down to 6.5 K. The
spectra rather show an asymmetric behavior across and a finite density of
states at the Fermi level ($E_F$) resembling those taken in the normal state of
bulk FeSe.
