  Recent years have seen a burgeoning interest in using pulsar timing arrays
(PTAs) as gravitational-wave (GW) detectors. To date, that interest has focused
mainly on three particularly promising source types: supermassive--black-hole
binaries, cosmic strings, and the stochastic background from early-Universe
phase transitions. In this paper, by contrast, our aim is to investigate the
PTA potential for discovering unanticipated sources. We derive significant
constraints on the available discovery space based solely on energetic and
statistical considerations: we show that a PTA detection of GWs at frequencies
above ~3.e-5 Hz would either be an extraordinary coincidence or violate
"cherished beliefs;" we show that for PTAs GW memory can be more detectable
than direct GWs, and that, as we consider events at ever higher redshift, the
memory effect increasingly dominates an event's total signal-to-noise ratio.
The paper includes also a simple analysis of the effects of pulsar red noise in
PTA searches, and a demonstration that the effects of periodic GWs in the 10^-8
-- 10^-4.5 Hz band would not be degenerate with small errors in standard pulsar
parameters (except in a few narrow bands).
