Why this is O(2ⁿ)
Why O(2ⁿ). A set of n elements has 2ⁿ subsets — we have to *produce* every one, so we can't beat 2ⁿ work just to write the output. The recursion doubles the result list at every level.
Lower bound. Even an optimal subset-generator can't be faster than O(2ⁿ) — there are 2ⁿ subsets and producing each one is a separate output. The complexity is bounded *from below* by the output size.
How to recognize exponential complexity in the wild
O(2ⁿ) is the cost of exploring every subset or every yes/no branch without pruning. Naive recursive Fibonacci, brute-force subset-sum, generating the power set.
The eight rungs of the ladder
Big-O classifies the asymptotic growth of a function, not the wall-clock time. The ladder Bugdle uses has eight rungs ordered from cheapest to most ruinous: O(1), O(log n), O(n), O(n log n), O(n²), O(n³), O(2ⁿ), O(n!). The puzzle's answer sits at rung 7 of 8. The four-guess budget plus higher/lower hints means the puzzle is solvable in at most ⌈log₂(8)⌉ = 3 perfectly-read guesses; the fourth attempt absorbs misreads of the snippet.
Common confusables
Nested loops aren't automatically quadratic — the inner loop has to range over n, not a constant. A loop that always runs ten iterations is still O(1) work per outer iteration. Similarly, a recursive function isn't automatically exponential just because it calls itself twice — if the recursion has overlapping subproblems and you memoise, you collapse back to polynomial. Always ask: what does n really mean here, and how many distinct subproblems are there?
External reference: Big O notation — Wikipedia.