Is there an overhead when nesting functions in Python?

The code object is pre-compiled so that part has no overhead. The function object gets built on every invocation — it binds the function name to the code object, records default variables, etc.

Executive summary: It’s not free.

>>> from dis import dis
>>> def foo():
        def bar():
                pass
        return bar

>>> dis(foo)
  2           0 LOAD_CONST               1 (<code object bar at 0x1017e2b30, file "<pyshell#5>", line 2>)
              3 MAKE_FUNCTION            0
              6 STORE_FAST               0 (bar)

  4           9 LOAD_FAST                0 (bar)
             12 RETURN_VALUE 

There is an impact, but in most situations it is so small that you shouldn’t worry about it – most non-trivial applications probably already have performance bottlenecks whose impacts are several orders of magnitude larger than this one. Worry instead about the readability and reusability of the code.

Here some code that compares the performance of redefining a function each time through a loop to reusing a predefined function instead.

import gc
from datetime import datetime

class StopWatch:
     def __init__(self, name):
         self.name = name

     def __enter__(self):
         gc.collect()
         self.start = datetime.now()

     def __exit__(self, type, value, traceback):
         elapsed = datetime.now()-self.start
         print '** Test "%s" took %s **' % (self.name, elapsed)

def foo():
     def bar():
          pass
     return bar

def bar2():
    pass

def foo2():
    return bar2

num_iterations = 1000000

with StopWatch('FunctionDefinedEachTime') as sw:
    result_foo = [foo() for i in range(num_iterations)]

with StopWatch('FunctionDefinedOnce') as sw:
    result_foo2 = [foo2() for i in range(num_iterations)]

When I run this in Python 2.7 on my Macbook Air running OS X Lion I get:

** Test "FunctionDefinedEachTime" took 0:00:01.138531 **
** Test "FunctionDefinedOnce" took 0:00:00.270347 **

The other answers are great and really answer the question well. I wanted to add that most inner functions can be avoided in python using for loops, generating functions, etc.

Consider the following Example:

def foo():
    # I need to execute a function on two sets of arguments:
    argSet1 = (1, 3, 5, 7)
    argSet2 = (2, 4, 6, 8)

    # A Function could be executed on each set of args
    def bar(arg1, arg2, arg3, arg4):
        return (arg1 + arg2 + arg3 + arg4)

    total = 0
    for argSet in [argSet1, argSet2]:
      total += bar(*argSet)
    print( total )

    # Or a loop could be used on the argument sets
    total = 0
    for arg1, arg2, arg3, arg4 in [argSet1, argSet2]:
        total += arg1 + arg2 + arg3 + arg4
    print( total )

This example is a little goofy, but I hope you can see my point nonetheless. Inner functions are often not needed.

I was curious about this too, so I decided to figure out how much overhead this incurred.
TL;DR, the answer is not much.

Python 3.5.2 (default, Nov 23 2017, 16:37:01) 
[GCC 5.4.0 20160609] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> from timeit import timeit
>>> def subfunc():
...     pass
... 
>>> def no_inner():
...     return subfunc()
... 
>>> def with_inner():
...     def s():
...         pass
...     return s()
... 
>>> timeit('[no_inner() for _ in range(1000000)]', setup='from __main__     import no_inner', number=1)
0.22971350199986773
>>> timeit('[with_inner() for _ in range(1000000)]', setup='from __main__ import with_inner', number=1)
0.2847519510000893

My instinct was to look at percents (with_inner is 24% slower), but that number is misleading in this case, since we’ll never actually just return the value of an inner function from an outer function, especially with functions that don’t actually do anything.
After making that mistake, I decided to compare it to other common things, to see when this does and does not matter:

    >>> def no_inner():
    ...     a = {}
    ...     return subfunc()
    ... 
    >>> timeit('[no_inner() for _ in range(1000000)]', setup='from __main__ import no_inner', number=1)
    0.3099582109998664

Looking at this, we can see that it takes less time than creating an empty dict (the fast way), so if you’re doing anything non-trivial, this probably does not matter at all.

Yes. This enables closures, as well as function factories.

A closure causes the inner function to remember the state of its environment when called.

def generate_power(number):

    # Define the inner function ...
    def nth_power(power):
        return number ** power

    return nth_power

Example

>>> raise_two = generate_power(2)
>>> raise_three = generate_power(3)

>>> print(raise_two(3))
8
>>> print(raise_three(5))
243
"""