That's a wonderful question, but I think there's a fundamental confusion here about two possible roles of the rational/trigonometric/elliptic trichotomy -- the one asked in the question and the one that leads to elliptic quantum groups -- which are in some sense "Fourier dual". (Everything I understand about this I learned talking to Tom Nevins.) For example the R/T/E trichotomy in R-matrices corresponds in quantum group world to the trichotomy Yangians/quantum affine algebras/elliptic quantum groups, not to group/quantum group/elliptic quantum group, or in nonquantum world to the trichotomy "Lie algebra/Lie group/elliptic group", with no quantums around. Not that there's an independent notion of an "elliptic group", but you can define a lot about it in terms of moduli of bundles on an elliptic curve. What the question asks is to fill in the trichotomy group/quantum group/"elliptically quantum" group..

Perhaps the easiest setting to see these two roles is in the study of many-body systems, eg the Calogero-Moser systems (or equivalently of meromorphic solutions of the KP and Toda hierarchies). These are completely integrable hamiltonian systems describing the motion of particles in the line. Initially it looks like they come in three flavors - rational, trigonometric and elliptic - labeled by whether the dependence of the potential on positions is rational, periodic or doubly periodic. This trichotomy is nicely explained by the trichotomy in R-matrices or in one-dimensional algebraic groups over $C$ or in irreducible genus one curves (Weierstrass cubics). The phase space can be described in terms of a cotangent bundle to a configuration space of points (on $C$, $C^\times$ or an elliptic curve), and the corresponding quantum systems can be described in terms of differential operators on the corresponding configuration spaces.
In representation theory this trichotomy appears in studying three versions of the loop algebra - current algebras from $C$, $C^\times$ or $E$ (the latter needs to be interpreted more sophisticatedly).One can (as I mentioned) invent something you call an "elliptic group" by studying G-bundles on an elliptic curve, in such a way that if your elliptic curve acquires a node you get the usual group, and if it acquires a cusp you get the Lie algebra.. but again this is not the question.

I claim this R/T/E trichotomy is naturally identified with the one in the above answers, but "linearly independent" (in fact Fourier dual) to the one in the question. In integrable systems world this is expressed as follows: there's a deformation of the Calogero-Moser particle systems (called Ruijsenaars or Ruijsenaars-Schneider or Macdonald) in which we change the rational dependence on MOMENTUM to trigonometric dependence on momentum. This corresponds to changing from the linear Poisson structure on the cotangent bundle to configuration space to a quadratic one on the multiplicative cotangent bundle to config space (replace all $C$'s by $C^\times$'s). When we quantize this quadratic Poisson bracket we get DIFFERENCE rather than differential operators, and a relation to quantum groups rather than groups as in the CM case.

Integrable systems people (eg Nekrasov, Gorsky and collaborators) like to express this by a 3 x 3 square : we can havce R/T/E dependence on position as above, or R/T/E dependence on momenta.. except no one has a good definition (AFAIK) of what elliptic dependence on momenta MEANS - except via Fourier transform, exchanging position and momenta - so you can define elliptic momenta/rational positions eg by switching the order..

In any case the row "rational momenta" relates to (loop) groups, "trig momenta" relates to quantum groups. So the question is what is the elliptic version?
There are various hints from string theory (see works of Nekrasov eg--- the rational row relates to SUSY 4dimensional gauge theory via the Seiberg-Witten solution, the trigonometric row to 5dimensional gauge theory, and the elliptic row should come from the mysterious "5-brane theory" or "6-dimensional (0,2) CFT")...

But maybe the most concrete answer I can give is motivated by Nakajima's work (see eg his ICM). You can realize representations of (loop) groups (of simple groups of type ADE) in equivariant cohomology of quiver varieties.
If you replace equivariant cohomology by equivariant K-theory, you find the representation theory of quantum (affine) algebras. So this gives a natural place to look for the elliptic analogue --- try to understand the equivariant elliptic cohomology of quiver varieties! such ideas were put forth by Grojnowski, Ginzburg-Kapranov-Vasserot, and others but a good theory of equivariant elliptic cohomology was only developed fairly recently by Lurie, so one can contemplate such questions anew.

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is broken. Possibly it is meant to point to the following article:Schlosser, Michael,Elliptic enumeration of nonintersecting lattice paths, J. Comb. Theory, Ser. A 114, No. 3, 505–521 (2007). Zbl 1119.05012. $\endgroup$