The ability of an observer to keep track of things is not relevant, and even completely computable things like cellular automata can be chaotic [1]. The common definition of chaotic dynamics [2] requires the system to be sensitive to initial conditions, mix the state space, and have dense periodic orbits.

[1] https://en.wikipedia.org/wiki/Rule_30#Chaos.

[2] https://en.wikipedia.org/wiki/Chaos_theory#Chaotic_dynamics