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 events 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).
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