  In this paper we prove the existence of a new type of relaxation oscillation
occurring in a one-block Burridge-Knopoff model with Ruina rate-and-state
friction law. In the relevant parameter regime, the system is slow-fast with
two slow variables and one fast. The oscillation is special for several
reasons: Firstly, its singular limit is unbounded, the amplitude of the cycle
growing like $\log \epsilon^{-1}$ as $\epsilon\rightarrow 0$. As a consequence
of this estimate, the unboundedness of the cycle cannot be captured by a simple
$\epsilon$-dependent scaling of the variables, see e.g. \cite{Gucwa2009783}. We
therefore obtain its limit on the Poincar\'e sphere. Here we find that the
singular limit consists of a slow part on an attracting critical manifold, and
a fast part on the equator (i.e. at $\infty$) of the Poincar\'e sphere, which
includes motion along a center manifold. The reduced flow on this center
manifold runs out along the manifold's boundary, in a special way, leading to a
complex return to the slow manifold. We prove the existence of the limit cycle
by showing that a return map is a contraction. The main technical difficulty in
this part is due to the fact that the critical manifold loses hyperbolicity at
an exponential rate at infinity. We therefore use the method in
\cite{kristiansen2017a}, applying the standard blowup technique in an extended
phase space. In this way we identify a singular cycle, consisting of $12$
pieces, all with desirable hyperbolicity properties, that enables the
perturbation into an actual limit cycle for $0<\epsilon\ll 1$.
