Determine the depth of an array

K, 4 bytes

#,/\

In K, ,/ will join all the elements of a list. The common idiom ,// iterates to a fixed point, flattening an arbitrarily nested list completely. ,/\ will iterate to a fixed point in a similar way, but gather a list of intermediate results. By counting how many intermediate results we visit before reaching the fixed point (#), we get the answer we want: the maximum nesting depth.

"Count of join over fixed-point scan".

In action:

 (#,/\)'(,1
        1 2 3
        ,1 2 3
        (3;(3;,3;3);3)
        ,((,1;2);(3;,4)))
1 1 2 3 4

Retina, 10

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+`\w|}{

{

Here the input format is a bit contrived - _ characters are used for list separators, so an input would look like this {1_{{2_3_{{4}_5}_6_{7_8}}_9_{10_{{{11}}}}_12_13}_14}

• Stage 1 - repeatedly remove }{ and all other \w characters. This has the effect of a) making all lists at all levels consist of only one element and b) removing all non-list-structural characters.
• Stage 2 - count remaining {. This gives the deepest level of nesting.

Try it online.

---

If that's too much of a stretch, then the previous answer was:

Retina, 13

Assumes lists are contained in curly braces {}.

+`[^}{]|}{

{

Try it online.

Python 2, 33 bytes

f=lambda l:l>{}and-~max(map(f,l))

Recursively defines the depth by saying the depth of a number is 0, and the depth of a list is one more than the maximum depth of its elements. Number vs list is checked by comparing to the empty dictionary {}, which falls above numbers but below lists on Python 2's arbitrary ordering of built-in types.