  We report our ongoing search for extremely inverted spectrum compact radio
galaxies, for which the defining feature in the radio spectrum is not the
spectral peak, but instead the slope of the spectrum (alpha) in the
high-opacity (i.e., lower frequency) part of the radio spectrum. Specifically,
our focus is on the spectral regime with spectral index, alpha >+2.5. The
motivation for our study is, firstly, extragalactic sources with such extreme
spectral index are extremely rare, because of the unavailability of right
combination of sensitivity and resolution over a range of low frequencies. The
second reason is more physically motivated, since alpha = +2.5 is the maximum
slope theoretically possible for a standard radio source emitting synchrotron
radiation. Therefore such sources could be the test-bed for some already
proposed alternative scenarios for synchrotron self-absorption (SSA), like the
free-free absorption (FFA) highlighting the importance of jet-ISM interaction
in the radio galaxy evolution.
