[...] almost all FPGA implementations must have a
clock of some sort. 
Oh!  That's a very important thing I hadn't known before.  I'd been
assuming an FPGA was, well, a big chipful of gates which you wired up
however you wanted, and which then behaved like, well, the bunch of
gates you've wired them up as; whether any of the signals those gates
processed could be considered a clock would be irrelevant.
I'm glad to have this mental model corrected _before_ I did anything
based on it.  That's a pretty crippling limitation for many purposes.
  Not every design employing logic is a CPU and not all
employ clocked
 logic. 
Quite.
I have a boardful of TTL which I designed and built to allow a parallel
port to control (via relays, driven off discrete transistors controlled
by logic levels) a bunch of mains outlets.  There are a few signals
which could sort of be seen as clocks - for example, one of them drives
the clock input to a shift register - but no global clock that applies
to the whole circuit.  Would this be unrepresentable in an FPGA - or,
rather, would it have to be redesigned to operate with a global clock
to be representable in an FPGA?  You make it sound like it.
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