Long used in its simplest form in mathematics, functional iteration is an elegant way to represent repeated operations. The Wolfram Language's symbolic architecture makes powerful general forms of functional iteration immediately accessible.

NestList — successively nest a function: {x,f[x],f[f[x]],f[f[f[x]]],…}

Nest — give the result of nesting a function: f[f[f[x]]] etc.

NestGraph — give the graph of nesting a function

FoldList — successively fold in a list of values: {x,f[x,1],f[f[x,1],2],f[f[f[x,1],2],3],…}

Fold — give the result of folding in a list of values: f[f[f[x,1],2],3] etc.

SequenceFold , SequenceFoldList — fold allowing sequences of previous values

FoldPair , FoldPairList — give a result and maintain a state at each step (e.g. quotient-remainder)

FoldWhile , FoldWhileList — fold while a condition is satisfied

FixedPoint , FixedPointList — nest until a fixed point is reached

NestWhile , NestWhileList — nest while a condition is satisfied

TakeWhile — take from a list while a condition is satisfied

LengthWhile — the length while a condition is satisfied