Maybe this wouldn't be my first choice, but I still think it's worth being on the list: Gödel's constructible universe $L$.

I would argue that it is intricate because it can serve as a model for "all of mathematics" (i.e., ZFC), furthermore answering many combinatorial questions left open by ZFC alone. Even though most(?) set theorists will probably argue that it is not "the" true model giving the right answer to these questions, it is still undoubtedly a rich and complex structure, moreover one in which the axiom of choice and the continuum hypothesis are not only true but "explained".

But it is also beautiful because of its connections with higher computability theory (e.g., the sets of integers constructed at the level $\omega_1^{\mathrm{CK}}$ of the constructible hierarchy, where $\omega_1^{\mathrm{CK}}$ is the smallest nonrecursive ordinal, are exactly the hyperarithmetical sets, i.e., the (lightface) $\Delta^1_1$ sets of the analytic hierarchy), and, in a related manner, because of Jensen's results on the "fine structure" of $L$. In a very intuitive way, I'd say that $L$ consists of sets that are ultimately "computable" (iterating the Turing jump as far as it can be), a perfectly regular construction that prohibits any randomness.

So even if set theorists are unhappy with $L$ because it forbids really large cardinals, and even if they try to construct something better (the core model), I argue that Gödel's original $L$ is still something immensely intricate and beautiful.