  We report the first extragalactic detection of the higher-order millimeter
hydrogen recombination lines ($\Delta n>2$). The $\gamma$-, $\epsilon$-, and
$\eta$-transitions have been detected toward the millimeter continuum source
N105-1A in the star-forming region N105 in the Large Magellanic Cloud (LMC)
with the Atacama Large Millimeter/submillimeter Array (ALMA). We use the
H40$\alpha$ line, the brightest of the detected recombination lines
(H40$\alpha$, H36$\beta$, H50$\beta$, H41$\gamma$, H57$\gamma$, H49$\epsilon$,
H53$\eta$, and H54$\eta$), and/or the 3 mm free-free continuum emission to
determine the physical parameters of N105-1A (the electron temperature,
emission measure, electron density, and size) and study ionized gas kinematics.
We compare the physical properties of N105-1A to a large sample of Galactic
compact and ultracompact (UC) H II regions and conclude that N105-1A is similar
to the most luminous ($L>10^5$ $L_{\odot}$) UC H II regions in the Galaxy.
N105-1A is ionized by an O5.5 V star, it is deeply embedded in its natal
molecular clump, and likely associated with a (proto)cluster. We incorporate
high-resolution molecular line data including CS, SO, SO$_2$, and CH$_3$OH
($\sim$0.12 pc), and HCO$^{+}$ and CO ($\sim$0.087 pc) to explore the molecular
environment of N105-1A. Based on the CO data, we find evidence for a
cloud-cloud collision that likely triggered star formation in the region. We
find no clear outflow signatures, but the presence of filaments and streamers
indicates on-going accretion onto the clump hosting the UC H II region. Sulfur
chemistry in N105-1A is consistent with the accretion shock model predictions.
