  Axion-like particles (ALPs) are a well-motivated dark matter candidate that
solve some of the problems in the clustering of large scale structure in
cosmology. ALPs are often described by a simplified quadratic potential to
specify the dynamics of the axion field, and are included in cosmological
analysis codes using a modified fluid prescription. In this paper we consider
the extreme axion: a version of the axion with a high initial field angle that
produces an enhancement (rather than a suppression) of structure on small
scales around the Jeans length, which can be probed by measurements of
clustering such as the eBOSS DR14 Ly-$\alpha$ forest. We present a novel method
of modeling the extreme axion as a cosmological fluid, combining the
Generalized Dark Matter model with the effective fluid approach presented in
the \texttt{axionCAMB} software, as well as implementing a series of
computational innovations to efficiently simulate the extreme axions. We find
that for axion masses between $10^{-23} \mathrm{\ eV} \lesssim m_\mathrm{ax}
\lesssim 10^{-22.5} \mathrm{\ eV}$, constraints on the axion fraction imposed
by the eBOSS DR14 Ly-$\alpha$ forest can be significantly weakened by allowing
them to be in the form of extreme axions with a starting angle between $\pi -
10^{-1} \lesssim \theta_i \lesssim \pi - 10^{-2}$. This work motivates and
enables a more robust hydrodynamical analysis of extreme axions in order to
compare them to high-resolution Ly-$\alpha$ forest data in the future.
