Do algebraic structures with more than two operations (maybe even infinitely many) exist?

If I have my history correct, they were originally invented to study loop spaces.

Yep. The story is a bit long, but the crux of the matter is that if X is a topological space, then ΩX (the loop space on X) has a multiplication, concatenation of loops, which is associative only up to homotopy. It turns out that then C_*(X), the singular chains complex of X, has the structure of an A_∞ algebra, in other words, a strongly homotopy associative algebra.
Another thing: the celebrated recognition principle says that if a topological space can be given an A_∞ structure (for topological spaces it's a different structure, that uses the little intervals operad), then under a technical condition X is actually homotopy equivalent to a loop space. It's pretty much amazing.

But as you see, an A_∞ algebra has morally only one "operation": the product. The rest is extra data encoding how the product is homotopy associative. There exist higher structures, called E_n algebras (n is an integer), that are morally made of n different products. They are all homotopy associative, and they are compatible in some sense. In particular E_1 = A_∞. If they were all really associative, then the Eckmann–Hilton argument would say that in fact, they are all equal and all commutative! But in general they are all only equal up to homotopy, and commutative up to homotopy. The bigger n is, the "more commutative" the structure is. In fact one can take n = ∞, and then an E_∞ is the next best thing to a commutative algebra. There is also a recognition principle (Boardmann–Vogt, May) for E_n algebras and loop spaces: an n-fold loop space Ωn(X) is an E_n algebra, and conversely under technical conditions an E_n-algebra is homotopy equivalent to an n-fold loop space.

If all this interested you, I suggest you read What is... an operad? by Jim Stasheff. It's only two (very dense though) pages long, and it's very interesting. It requires some background in algebraic topology to understand fully.

/r/math Thread Parent