Iterated aliquot sums
Related to case (b) above is a system which repeats the replacement n->Apply[Plus, Divisors[n]]-n or equivalently n->DivisorSigma[1, n]-n. The fixed points of this procedure are the perfect numbers (see above). Other numbers usually evolve to perfect numbers, or to short repetitive sequences of numbers. But if one starts, for example, with the number 276, then the picture below shows the number of base 10 digits in the value obtained at each step.
After 500 steps, the value is the 53-digit number
The question of whether such values can increase forever was considered by Eugène Catalan in 1887, and has remained unresolved since.