This precisely mean that language without second stack (more or less) is not Turing complete.

But, if your language can store lists on a stack, you're Turing complete again! One example if a Joy language, where you can have quotation on stack, assemble them and decompose back.

Most of the time you store there various intermediate values. It depends on the nature of language. For example, Forth variants often store loop counters on retain stack.