  The pursuit of discovery in particle physics has always required study at the
highest possible energies. Multi-TeV muon colliders, which have important
advantages, offer unprecedented potential for investigating new physics beyond
the Standard Model. We extensively assume a range of center-of-mass energies
from 3 to 30 TeV and a range of integrated luminosities from 1 to 90 ab$^{-1}$
for future multi-TeV muon colliders. We perform a phenomenological study of the
anomalous $WW\gamma$ couplings via the process $\mu^+
\mu^-\,\rightarrow\,\mu^+\gamma^* \mu^-\,\rightarrow\,\mu^+\ell^-\nu_\ell
\bar{\nu}_\ell$ using a model-independent analysis in the effective field
theory framework at multi-TeV muon colliders. The sensitivity estimates
obtained for the anomalous $WW\gamma$ couplings at systematic uncertainties of
$\delta_{sys}=0$, $3$, $5\%$ are compared with the experimental results. Our
best sensitivity limits at 95$\%$ confidence level for $c_{WWW}/{\Lambda^2}$,
$c_{\widetilde{W}WW}/\Lambda^2$, $c_{B}/{\Lambda^2}$, $c_{W}/{\Lambda^2}$ and
$c_{\widetilde{W}}/\Lambda^2$ couplings are $[-0.030; 0.025]\,\text{TeV}^{-2}$,
$[-0.014; 0.014]\,\text{TeV}^{-2}$, $[-0.185; 0.187]\,\text{TeV}^{-2}$,
$[-0.187; 0.189]\,\text{TeV}^{-2}$ and $[-1.177; 1.097]\,\text{TeV}^{-2}$ at
the multi-TeV muon collider with center-of-mass energy of 30 TeV and integrated
luminosity of 90 ab$^{-1}$, respectively.
