  We consider the cosmological consequences if a small fraction ($f\lesssim
0.1$) of the dark matter is ultra-strongly self-interacting, with an elastic
self-interaction cross-section per unit mass $\sigma\gg1\ \mathrm{cm^{2}/g}$.
This possibility evades all current constraints that assume that the
self-interacting component makes up the majority of the dark matter.
Nevertheless, even a small fraction of ultra-strongly self-interacting dark
matter (uSIDM) can have observable consequences on astrophysical scales. In
particular, the uSIDM subcomponent can undergo gravothermal collapse and form
seed black holes in the center of a halo. These seed black holes, which form
within several hundred halo interaction times, contain a few percent of the
total uSIDM mass in the halo. For reasonable values of $\sigma f$, these black
holes can form at high enough redshifts to grow to $\sim10^9 M_\odot$ quasars
by $z \gtrsim 6$, alleviating tension within the standard $\Lambda$CDM
cosmology. The ubiquitous formation of central black holes in halos could also
create cores in dwarf galaxies by ejecting matter during binary black hole
mergers, potentially resolving the "too big to fail" problem.
