Apply f to x from 0 to 4 times:

The function to nest can be a pure function:

Nesting a function can build a formula:

Nesting can return a single number:

The nested function can operate on a list:

Apply the operator form to different starting expressions:

Use Throw to exit a Nest:

To nest a function of more than one argument, the arguments can be put into a list:

Continued fraction:

Power tower:

Growth of annually compounded capital in 10 years:

Newton iterations for :

Iterated string replacements:

Consecutive pairs of Fibonacci numbers:

Functional composition for higher-order Newton iteration (for ):

Generate a bifurcation diagram for an iterated logistic map:

Nest gives the last element of NestList:

Nest is effectively Composition using the same function multiple times:

Use RSolve to symbolically compute Nest operations:

Different length compositions or nests give the same result:

FixedPoint automatically goes on until the result no longer changes:

NestWhile goes on while a condition is true:

Fold automatically inserts second arguments from a list:

Binary tree:

Gray codes of length 4: