How to create a nested list of variable depth with indexed elements

You can try Array:

Array[c, {3, 3, 3, 3}]

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arraydepth = 3;
Array[c, ConstantArray[3, arraydepth]]

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To elaborate on b.gates.you.know.what's comment, this is a scalable version equivalent to kglr's solution:

Outer[c, Sequence @@ ConstantArray[{1, 2, 3}, 4]] === Array[c, {3, 3, 3, 3}]

True

{1, 2, 3} can be a list of any elements.

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