Turing machines without time steps? In "More Is Different," an article about reductionism in science, the author makes the following off hand remark near the end:


  I find that at least one further phenomenon seems to be identifiable and either universal or remarkably common, namely, ordering (regularity or periodicity) in the time dimension... [It] is noteworthy that all computing machines use temporal pulsing.


As far as Turing machines go, this is true since there is a fundamental "time step" during which a symbol is read, the state is changed, and the read/write head moved.  But this got me wondering: are there any non-Turing computing proposals without this fundamental time step?
There are various models of analog computation, which embody computations which neither involve discrete states nor times. One of the more studied models is what might be called "Shannon machines" (after Claude Shannon): the General Purpose Analog Computer (GPAC).

Shannon's original formulation of GPAC could not compute certain functions, including the Gamma function. However, a 2007 paper cited in that Wikipedia article provides a reasonable reinterpretation of GPAC which turns out to be more powerful, enough so that the authors claim that "the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals."

Claude Shannon is well-known as a pioneer in digital computer design and particularly for his foundational work on information theory, but his genius and sense of whimsy led him to extraordinary achievements in other areas, and analog computation remained an interest throughout his life. One example is his mathematical analysis of juggling, including the construction of a prototype juggling robot. (See here for more information and a brief video including the robot.)