  We study the axion field range and low energy couplings in models with
St\"uckelberg mixing between axions and $U(1)$ gauge bosons. It is noted that
the gauge-invariant axion combination $\xi$ in the model is periodic $modulo$
an appropriate shift of gauge-variant axions eaten by the massive $U(1)$ gauge
bosons, which in some cases makes the connection between the field range and
the low energy couplings less transparent. We derive the field range of $\xi$
for generic forms of the axion kinetic metric and $U(1)$ charges, and identify
the field basis for which all non-derivative couplings of $\xi$ are quantized
in a manner manifestly consistent with the periodicity of $\xi$. Generically
St\"uckelberg mixing reduces the axion field range. In particular, the mixings
between $N$ axions and $(N-1)$ $U(1)$ gauge bosons typically result in an
exponentially reduced field range $M_\xi ={\cal O}\left({k^{-(N-1)}
f}/{\sqrt{N!}}\right)$ for the residual gauge-invariant axion $\xi$ in the
limit $N\gg 1$, where $f$ and $k$ denote the typical decay constant and the
root mean square of the $U(1)$ gauge charges of the original $N$ axions. Using
simple examples, we study also the reparameterization-invariant physical
quantities such as the axion effective potential and 1PI couplings to gauge
bosons, which are determined by the reparameterization-dependent axion
couplings in the model.
