The study of sets with an arbitrary number of operations is called universal algebra.
Also universal algebra isn't limited to binary operations but studies operations of any arity: nullary, unary, binary, ternary, ... , n-ary.
(Universal algebra does however typically restrict itself to axioms that are defined by equations which means fields are excluded from this way of studying algebra.)
Beware however that operations that are ostensibly independent might not be. See for example the earlier mathoverflow question: Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
In a ring the distributive law connects the addition and multiplication operations, so that they cannot in a sense be independent, but there is nothing to stop a structure of n operations from being consistent if none of the axioms connect any of the operations to each other. Like Gerhard Paseman comments about, it depends what you mean by independent.
Questions along similar lines to the linked question about universal operations could be asked about any given universal-algebraic variety - given a variety in which every operation is connected to the other operations by axioms, then to what extent can the number of operations be reduced and still define the same structureclass of structures. For questions about reducing the number of operations or axioms see https://www.cs.unm.edu/~mccune/projects/gtsax/
P.S. I'll mention here as a curiosity the topic of n,m-operations. That is operations that not only have a domain-arity but also a co-domain arity. e.g. $*:G \times G \times G \rightarrow G \times G$. Concepts such as associativity can be generalized to n,m-operations but n,m-operations haven't been studied much in universal algebra.
