The Gaussian free field (GFF) is an interesting object in its own right, and it describes various other phenomena -- from height models (for instance, those related to dimers) to the free bosonic field (which plays a key role in constructions of a number of conformal field theories (CFTs); for instance, the Liouville theory and its cousins). Let us first consider the GFF on planar simply connected domains with piecewise constant Dirichlet boundary data. The fractal geometry encoded by its level sets can be described via crossing events. These, in turn, can be solved by certain ``conformal blocks'' of primary fields in a CFT with central charge $c=1$. In particular, the geometry of the level-set percolation for the (both continuum and metric graph) GFF has a CFT description in terms of these conformal blocks (which are linearly independent and solve the Belavin-Polyakov-Zamolodchikov (BPZ) PDEs of arbitrary orders). Moreover, (not surprisingly) the theory is very integrable -- e.g., in the sense that the conformal blocks admit explicit formulas, involving Specht polynomials and their generalizations, and witnessing a rich algebraic content: they carry representations of symmetric groups and planar algebras, and can be expressed using isomonodromic tau-functions. Let us then consider boundary-to-boundary pattern probabilities in the double-dimer model in planar simply connected domains. In this case, the discrete explicit formulas can be written in terms of products of determinants (in the spirit of bosonization), which persists in the scaling limit. One can check that these formulas coincide with those for the GFF mentioned above. Analogous formulas are expected, and in some cases proven, to hold for multi-dimer webs of higher degree, as well as the level sets of their purported height functions. (These models should have central charge $c=2,3,\ldots$ and, while being planar, might secretly be something beyond.)