Turing machines without time steps?
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.)