  We study Higgs boson production at $\mu^+ \mu^+$ colliders at high energy.
Since both initial-state particles are positively charged, there is no $W$
boson fusion at the leading order, as it requires a $W^+ W^-$ pair. However, we
find that the cross section of the higher-order, $\gamma$- and $Z$-mediated $W$
boson fusion process is large at high center-of-mass energies $\sqrt s$,
growing as $(\log s)^3$. This is in contrast to the $\log s$ behavior of the
leading-order $W$ boson fusion. Thus, even though it is a higher-order process,
the rate of Higgs boson production for 10 TeV energies at $\mu^+ \mu^+$
colliders with polarized beams can be as high as about half of the one at
$\mu^+ \mu^-$ colliders, assuming the same integrated luminosity. To calculate
the cross section of this process accurately, we carefully treat the collinear
emission of the photon in the intermediate state. The thereby obtained large
cross section furthermore shows the significance of Higgs production with an
extra $W$ boson in the final state also at $\mu^+ \mu^-$ and $e^+ e^-$
colliders.
