The eternity clock is a device to quiet what Douglas Adams called "The swishing sound of deadlines as they go by" (I have paraphrased this a fair bit). This makes it especially useful during software development. You tell it what the deadline is and start it running. It starts off by running at the same time as a normal clock, but slows down as the deadline approaches, so that the deadline is approached asymptotically, but never actually reached. Of course, if you view the eternity clock time as an estimate of progress actually made, it is less reassuring.
All we need for this is a function f(x) that has f'(x)=1 at 0 and approaches some limit A as x goes to infinity. The first one that came to mind was Ax/(A + x). The Javascript program below can be used to convert from inconvenient wall clock time to reassuring eternity clock time.