Apologies for the vague title and soft question. According to Etingof, Igor Frenkel once suggested that there are three "levels" to Lie theory, which I guess could be given the following names:

**No loops:**here we study a simple Lie algebra $\mathfrak{g}$, a Weyl group, a braid group, or a Hecke algebra, all of which have something to do with a Lie group $G$.**One loop:**here we study an affine Lie algebra $\widehat{\mathfrak{g}}$, a quantum group $U_q(\mathfrak{g})$, an affine Weyl group, an affine braid group, or an affine Hecke algebra, all of which I think have something to do with the loop group $LG$ of $G$.**Two loops:**here we study a double affine Lie algebra $\widehat{\widehat{\mathfrak{g}}}$, an affine quantum group $U_q(\widehat{\mathfrak{g}})$, an elliptic quantum group (whatever that means), a double affine or elliptic Weyl group, a double affine or elliptic braid group, or a double affine or elliptic Hecke algebra, all of which I think have something to do with the double loop group of $G$, or maybe more precisely the space of maps of some sort from an elliptic curve $E$ to $G$.

The suggestion is further that this pattern doesn't continue.

Why doesn't this pattern continue?

I asked around and got an answer that I interpreted as follows. The trichotomy above can be matched up to the trichotomy additive group $\mathbb{C}$ : multiplicative group $\mathbb{C}^{\times}$ : elliptic curve $E$. Here is one story about the match as I understand it, which is not very well.

- One-dimensional algebraic groups give rise to equivariant cohomology theories. The above theories give rise to equivariant cohomology, equivariant K-theory, and equivariant elliptic cohomology respectively.
- Roughly, $\text{Spec } H^{\bullet}_G(\text{pt}) \cong \mathfrak{g}/G \cong \text{Bun}_G(\mathbb{C})$, while $\text{Spec } K^{\bullet}_G(\text{pt}) \cong G/G \cong \text{Bun}_G(\mathbb{C}^{\times})$, and $\text{Spec } E^{\bullet}_G(\text{pt}) \cong \text{Bun}_G(E)$, where by $E^{\bullet}_G$ I mean the equivariant elliptic cohomology theory associated to the elliptic curve $E$.
- There is some yoga in geometric representation theory which I'm not all that familiar with involving building interesting algebras like group algebras of Weyl groups and Hecke algebras by computing the equivariant (co)homology or equivariant K-(co)homology of some varieties of interest, which has something to do with the construction of the affine and double affine Hecke algebras mentioned above.

Since we've run out of one-dimensional algebraic groups, that would be some reason to believe that the pattern doesn't continue. But nevertheless I don't have a good sense of what, if anything, prevents us from studying and saying interesting things about "triple affine Lie algebras," "triple affine Weyl groups," "triple affine Hecke algebras," etc. at least insofar as the triple loop group of a group seems perfectly well-defined. On the geometric side it seems like there's nothing stopping us studying $G$-bundles on higher dimensional varieties. On the cohomological side, cohomology, K-theory, and elliptic cohomology should optimistically just be the first three terms of an entire sequence of cohomology theories at higher chromatic levels, or from the perspective of the Stolz-Teichner program, defined in terms of higher-dimensional field theories...