Variant of showing every positive integer is a sum of sufficiently many $k$th powers of non-negative integers Is it known whether every positive integer can be expressed as the sum of a square of a non-negative integer, a cube of a non-negative integer, and so forth, up to some finite bound $k$ of a $k$th power of a non-negative integer?
 A: The answer is Yes.
Quoting from Richard K. Guy's famous book "Unsolved problems in Number Theory"
( Section C21 "Sum of higher powers" )

Brüdern proved that all sufficiently large $n$ can be represented as the sum of $17$ distinct powers, $x_1^2 + x_2^3 + \cdots + x_{17}^{18}$. Ford improves this to $15$ and
  later to $14$.

I looked up one of Ford's paper in the reference (an online copy can be found 
here )

Kevin B. Ford, The representation of numbers as sum of unlike powers, II,
J. Amer Math. Soc., ${\bf 9}(1995)$ 818-940; MR 97g 11111.

What has been proved does refer to sum of unlike powers of non-negative integers.
Let $N$ be the largest natural number such that
$$ N \not\in \left\{ n \in \mathbb{Z}_{+} : \exists x_i \in \mathbb{N} , n = \sum_{i=1}^{14} x_i^{i+1}\right\}$$
It is clear every positive number $n \le N$ can be represented as
$$ n = \sum_{i=1}^n 1^{i+1}$$
This implies every natural number $n$ can be represented as a sum of no more
than $\max(14,N)$ powers of non-negative integers.
