The typical functions I see with finite Lyapunov times are merely exponential; they only generate (e times larger) infinitesimal differences in final conditions from infinitesimal differences in initial conditions. This means that, in principle, it's possible to keep increasing the precision of initial conditions to keep extending the predictability of the modeled system out forever. Not in practice, but in principle.
I'm interested in looking at some functions where that is no longer possible, even in principle.
For a continuous phase-space flow $Φ$, infinitesimal changes to a state $x$ only beget infinitesimal changes to the evolved state $Φᵗ(x)$. That’s already the definition of continuity. So, what you want is equivalent to discontinuities in the phase-space flow.
For a dynamical system, your flow can at best be discontinuous at discrete points. For such a system, you can have what you want at the discontinuities. A very simple example is the bit-shift map, which is discontinuous at $\tfrac{1}{2}$: The flow maps $\tfrac{1}{2}$ to $1$ and $\tfrac{1}{2}+ε$ to $2ε$.
What you cannot have is that the flow is discontinuous everywhere, because then nobody (I am aware of) would call the result a dynamical system – or flow for that matter.