Various models generalising Turing's machine
have been investigated with constraints of finite time
(and possibly also space) being lifted. The most well known of these
is the Infinite Time Turing machine of Hamkins and Kidder,
but it is clear that many machines can be reconfigured
to go transfinite, such as register machines (Koepke et al), or more recently Blum-Shub-Smale Machines (see
Koepke-Seyfferth at this conference)
all with differing characteristics.
Indeed there would seem to be many possibilities of transcending
the machine inspiration and for the direct generalisation
of dynamical systems (broadly conceived) into the transfinite.
Probably this would result in varying degrees of
significance. In general one would expect the transfinite to be very different from the finite version but
nevertheless some leading to interesting mathematics.