How high would it go?

The set A is defined as

\(A=\left\{n|n=\overline{d_kd_{k-1}\cdots d_{1}},n=\sum_{i=1}^k d_i^k\right\}\)

In other words,

\(n\) is a k-digit number which is equal to the sum of the k-th powers of its digits.

Eg : \(153=1^3+5^3+3^3\)

What is the maximum possible number of digits that any number in the set can have?

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