  The BCS-BEC crossover from strongly overlapping Cooper pairs to
non-overlapping composite bosons in the strong coupling limit has been a
long-standing issue of interacting many-body fermion systems. Recently, FeSe
semimetal with hole and electron bands emerged as a high-$T_{\rm c}$
superconductor located in the BCS-BEC crossover regime, owing to its very small
Fermi energies. In FeSe, however, an ordinary BCS-like heat-capacity jump is
observed at $T_{\rm c}$, posing a fundamental question on the characteristics
of the BCS-BEC crossover. Here we report on high-resolution heat capacity,
magnetic torque, and scanning tunneling spectroscopy measurements in
FeSe$_{1-x}$S$_x$. Upon entering the tetragonal phase at $x>0.17$, where
nematic order is suppressed, $T_{\rm c}$ discontinuously decreases. In this
phase, highly non-mean-field behaviors consistent with BEC-like pairing are
found in the thermodynamic quantities with giant superconducting fluctuations
extending far above $T_{\rm c}$, implying the change of pairing nature.
Moreover, the pseudogap formation, which is expected in BCS-BEC crossover of
single-band superconductors, is not observed in the tunneling spectra. These
results illuminate highly unusual features of the superconducting states in the
crossover regime with multiband electronic structure and competing electronic
instabilities.
