Do chaotic systems exist that cannot be predicted even at the limit of inifinite precision initial conditions and infinite resources? I have a layman person's understanding of the theory of chaos, that seems to indicate that using finite-precision initial conditions and finite computing resources, chaotic systems cannot be predicted after a period of time.
My question is what happens in the limit of increasing the precision of initial conditions and resources to infinity: Does the system remain chaotic, or does the prediction window also diverge to infinity?
Specifically consider the following conditions:

*

*We have a chaotic system.


*We calculate the prediction time window  $t_\text{pred}(e,p,m,s)$ given a finite error margin $e$, for a finite precision of initial conditions $p$, and a computer with finite memory $m$ operating at a finite speed $s$.


*We calculate the same prediction time window $t_\text{pred}(e,p,m,s)$ when precision, memory, and speed diverge to infinity together (but $e$ remains finite).

*

*If for all chaotic systems the time window diverges to infinity, then the answer to this question is no.


*If any system is found where $t_\text{pred}$ may remain finite, then the answer to this question is yes.

Since this questions seems very far from being practical I will add a motivation: I feel the answer of this question has an important impact in theology. Namely if the answer is yes then that would logically preclude the possibility of a non-interventionist, all-knowing god (future included) who designed the universe with a purpose, because he/she wouldn’t be able to do these calculations even if he/she was infinitely powerful.
 A: A crucial property of chaotic systems is that they are deterministic: There is no element of randomness in the model. The initial conditions exactly determine the future of the system.
If I simulate a chaotic model with the same initial conditions¹ on a real computer twice, I obtain exactly the same result. This only differs from the true solution for my initial conditions due to the finite precision of floating-point arithmetics (and, as the system is chaotic, this difference can be large)². And of course, in the purely hypothetical case that I want to simulate an isolated real system for which I have an exact model, I have the problem that I cannot perfectly represent my real initial conditions as floating-point numbers.
If I have arbitrary precision and infinite computing resources available as well as perfect knowledge of initial conditions, I can predict a chaotic system perfectly by simply simulating it. For a discrete-time system, the only reasons I need infinite memory and computation speed are storing and working with arbitrary precision numbers³ (and of course if I want to go infinitely into the future). For a continuous-time system, there is another reason I need infinite computing speed, namely to perform the numerical integration with arbitrarily fine time steps.


¹ and the same rules of floating-point arithmetics
² for a continuous-time system, the inherent imprecision of numerical integration also adds an error
³ since I end up with infinitely many digits

