  We report results on the first matched-filtering search for binaries with
compact objects having large tidal deformabilities in the LIGO-Virgo
gravitational wave (GW) data. The tidal deformability of a body is quantified
by the ``Love number" $\Lambda \propto \hskip 1pt (r/m)^5$, where $r/m$ is the
body's (inverse) compactness. Due to its strong dependence on compactness, the
$\Lambda$ of larger-sized compact objects can easily be many orders of
magnitude greater than those of black holes and neutron stars, leaving phase
shifts which are sufficiently large for these binaries to be missed by binary
black hole (BBH) templated searches. In this paper, we conduct a search using
inspiral-only waveforms with zero spins but finite tides, with the search space
covering chirp masses $3 M_\odot < \mathcal{M} < 15 M_\odot$ and effective
tidal deformabilities $10^2 \lesssim \tilde{\Lambda} \lesssim 10^6$. We find no
statistically significant GW candidates. This null detection implies an upper
limit on the merger rate of such binaries in the range $[1-300] \hskip 2pt
\text{Gpc}^{-3} \text{year}^{-1}$, depending on $\mathcal{M}$ and
$\tilde{\Lambda}$. While our constraints are model agnostic, we discuss the
implications on beyond the Standard Model scenarios that give rise to boson
stars and superradiant clouds. Using inspiral-only waveforms we recover many of
the BBH signals which were previously identified with full
inspiral-merger-ringdown templates. We also constrain the Love number of black
holes to $\Lambda \lesssim 10^3$ at the 90\% credible interval. Our work is the
first-ever dedicated template-based search for compact objects that are not
only black holes and neutron stars. Additionally, our work demonstrates a novel
way of finding new physics in GW data, widening the scope of potential
discovery to previously unexplored parameter space.
