If there's one layer of clothes then you're correct, it O(1). On further thought deeper piles are not O(log N), but O(N). Once the number of items exceeds C it takes more than a single operation to retrieve an item from the bottom layer, and the number of operations is proportional to the number of layers, or N/C.

If you consider either picking an item up or moving it aside as a single operation, then retrieval from a single layer take 1 operation, and is O(1), but retrieval from the bottom of a two layer pile actually takes 3 operations (move the top item, retrieve the target item, replace the top item into the bottom layer, or you risk getting a deeper pile in one slot in the pathalogical case). Retrieval from the bottom of 3 layers takes 5 operations (move, move, take, replace, replace). in other words we have an O(1) process for taking the target item, and an O(N/C)=O(N) process for uncovering it in the first place, giving O(N) over all.

Your statement that "considering that the naximum amount of clothes is likely very small, it can be treated as O(1)." is true iff N<=C, which, I concede, is a likely scenario in any well managed laundry pile, hence comment about cache sizing.

I only put "if it's big" to help readers imagine a large pile and how it would become more inconvenient the larger the pile. Language is not as precise as math or programming. If you'd like you can imagine the sentence as saying something like "imagine a big pile. It's definitely not O(1)"

It is quite reasonable to limit the average case O to the most common inputs with the caveat that it can be worse otherwise. It's not like every arithmetic operation is counted as O(n) in Python for example as you aren't expected to use numbers larger than 2^32 often