Questions. Have the following kinds algebraic structures been considered in the abstract algebra literature etc.? If so, what are they really called? (I have used made-up terminology for the sake of the question.)
A woodland is a set equipped with an $n$-ary operation $f_n$ for each integer $n \geq 0$. (Observe that every woodland has a distinguished element corresponding to the case $n=0$).
A jungle is a woodland such that we can permute the arguments of $f_n$ willy-nilly. In particular, $(X,f_*)$ is a jungle iff for all integers $n \geq 0$ and all permutations $\pi$ of $\{0,\ldots,n-1\},$ it holds that $f_n(x_0,\ldots,x_{n-1}) = f_n(x_{\pi(0)},\ldots,x_{\pi(n-1)}).$
Motivation. The rooted trees of graph theory form a jungle in an obvious way, and this is (isomorphic to) the initial jungle. Similarly, the ordered rooted trees form a woodland in an obvious way; and, this is the initial woodland.
A "woodland" would be an algebra over the free nonsymmetric operad $W$ with one generating operation in each arity $n \ge 0$ and no relations. Similarly a "jungle" would be an algebra over the free symmetric operad $J$ with one generating operation in each arity $n \ge 0$ and $\Sigma_n$ acts trivially on the $n$th generating operation. Trees appear naturally because this is what operads are built on. It seems to me that the set of (resp. ordered) rooted trees is in fact the free $W$-algebra (resp. $J$-algebra) over the empty set (the initial set), which explains why they're initial in their category of algebras.
I don't believe these things have been studied on their own, as they have very little structure; consider that a magma is a woodland where all the $f_n$ are trivial for $n \neq 2$, and magmas aren't particularly studied. They're more of an intermediary technical tool.
One (rather silly) thing I can say is that if you consider a linear version of this and remove the operations in arity $0$ and $1$, then the first one is the ungraded underlying operad of the dg-operad $A_\infty$ controlling homotopy associative algebras (it would have $\operatorname{deg} f_n = n - 2$ and a differential $\partial(f_n)$ that I don't want to write down). I guess you can relate the second one to some shifted version of $L_\infty$. I doubt this is what you had in mind though.