  Standard exceptional points (EPs) are non-Hermitian degeneracies that occur
in open systems. At an EP, the Taylor series expansion becomes singular and
fails to converge -- a feature that was exploited for several applications.
Here, we theoretically introduce and experimentally demonstrate a new class of
parity-time symmetric systems [implemented using radio frequency (rf) circuits]
that combine EPs with another type of mathematical singularity associated with
the poles of complex functions. These nearly divergent exceptional points can
exhibit an unprecedentedly large eigenvalue bifurcation beyond those obtained
by standard EPs. Our results pave the way for building a new generation of
telemetering and sensing devices with superior performance.
