[nullc]

If your computational model requires that the algorithm reads the input then I don't see how it's possible. Even if above some size the algorithm does nothing but exits the time will still grow linearly with the input from reading.

You could imagine an algorithm that takes some astronomical time for size 1, less for 2, etc.. but for any finite time at size 1 there will be some size N where the reading time finally dominates the computation.