  For the first time, limits on double-beta processes in $^{74}$Se have been
obtained using a 400 cm$^3$ HPGe detector and an external source consisting of
natural selenium powder.
  At a confidence level of 90%, they are $1.9\times 10^{18}$ y for the
$\beta^+$EC$(0\nu + 2\nu)$ transition to the ground state, $7.7\times 10^{18}$
y for the ECEC($2\nu$) transition to the $2^+_1$ excited state in $^{74}$Ge
(595.8 keV), $1.1\times 10^{19}$ y for the ECEC($0\nu$) transition to the
$2^+_1$ excited state in $^{74}$Ge (595.8 keV) and $5.5\times 10^{18}$ y for
the ECEC($2\nu$) and ECEC($0\nu$) transitions to the $2^+_2$ excited state in
$^{74}$Ge (1204.2 keV). The last transition is discussed in association with a
possible enhancement of the decay rate, in this case by several orders of
magnitude, because the ECEC$(0\nu)$ process is nearly degenerate with an
excited state in the daughter nuclide. Prospects for investigating such
processes in future experiments are discussed.
