  LHAASO J2108+5157 is one of the few known unidentified Ultra-High-Energy
(UHE) gamma-ray sources with no Very-High-Energy (VHE) counterpart, recently
discovered by the LHAASO collaboration. We observed LHAASO J2108+5157 in the
X-ray band with XMM-Newton in 2021 for a total of 3.8 hours and at TeV energies
with the Large-Sized Telescope prototype (LST-1), yielding 49 hours of good
quality data. In addition, we analyzed 12 years of Fermi-LAT data, to better
constrain emission of its High-Energy (HE) counterpart 4FGL J2108.0+5155. We
found an excess (3.7 sigma) in the LST-1 data at energies E > 3 TeV. Further
analysis in the whole LST-1 energy range assuming a point-like source, resulted
in a hint (2.2 sigma) of hard emission which can be described with a single
power law with photon index Gamma = 1.6 +- 0.2 between 0.3 - 100 TeV. We did
not find any significant extended emission which could be related to a
Supernova Remnant (SNR) or Pulsar Wind Nebula (PWN) in the XMM-Newton data,
which puts strong constraints on possible synchrotron emission of relativistic
electrons. The LST-1 and LHAASO observations can be explained as inverse
Compton-dominated leptonic emission of relativistic electrons with a cutoff
energy of $100^{+70}_{-30}$ TeV. The low magnetic field in the source imposed
by the X-ray upper limits on synchrotron emission is compatible with a
hypothesis of a PWN or a TeV halo. The lack of a pulsar in the neighborhood of
the UHE source is a challenge to the PWN/TeV-halo scenario. The UHE gamma rays
can also be explained as $\pi^0$ decay-dominated hadronic emission due to
interaction of relativistic protons with one of the two known molecular clouds
in the direction of the source. The hard spectrum in the LST-1 band is
compatible with protons escaping a shock around a middle-aged SNR because of
their high low-energy cut-off.
