That is a definition. But mathematics is made of theorems. Are there interesting theorems using that definition? If there are, I am all ears, and that motivates me to study it. Otherwise, I'll just ditch it.

As Dan said, magmas are too general, too abstract, so they are uninteresting. To quote Whitehead: "it is the large generalization, limited by a happy particularity, which is the fruitful conception."

"Magma", as I had it explained to me once, is no more than a French word meaning "jumble". It's as good a word as any for a set with a random binary mapping with no rules imposed on it.

As an idea of how loose the structure imposed by a magma is, there's 1 group of order $5$ and $2,\!483,\!527,\!537,\!094,\!825$ magmas of order $5$ (up to isomorphism, as listed in the OEIS).

@PrimeMover, the word means that in French but also it means, well, magma: the hot liquid fluid that is inside the Earth and which is the basis of everything

From a structural point of view, I wholeheartedly agree with OP. It's nice to know the hierarchy of objects and leaving out something so basic (/fundamental/simple) always leaves a sour taste for me.

This seems to me to be (a) opinion-based and (b) wrong: the most fundamental algebraic structure is one with a single monadic function and this is very widely studied and discussed (usually under the name of a set with a successor function).

Perhaps there are magmas of interest because they are not also something more well-known. Pointing to a group and noting that it is also a magma doesn't excite me. If you show me a magma that seems to always 'avoid' having regular structure (i.e. something like pseudorandomness), that itself might be interesting to me.

Here is probably the original source of the image.

"A magma is perhaps the simplest thing you could explain, way simpler than groups:" << By that measure, a topological space is "simpler" than a metric space and a metric space is "simpler" than a normed vector space and a normed vector space is "simpler" than a finite-dimensional normed vector space. But the opposite is true in practice: topological spaces are more abstract than finite-dimensional normed vector spaces, which makes them much much more complicated.

@PrimeMover - as a French I have never heard magma being used as jumble. The etymology is Latin "thick liquid".

See this nice Question and Answer about the etymology of magma over at History of Science and Math.

@Stef I'd say this is the best answer given so far, perhaps you should post it as such.

@leftaroundabout I had begun writing an answer but I realised I was talking only about topological spaces and not at all about magmas. I actually don't have much to say about magmas.

@leftaroundabout Dan Uznanski's answer says the opposite - that magmas are so abstract that there is nothing to say about them. So maybe it's a bit like games: tictactoe has so simple rules that its strategies are trivial; the game of go has slightly more complicated rules, which makes the strategies immensely complex; and monopoly has even more rules, which makes the strategies much easier. Magma is tictactoe, topological spaces are the game of go, and finite-dimensional normed vector spaces are monopoly.

@Stef ...which is an even better answer!

@GiuseppeNegro Here's an example of a "theorem" regarding magmas: suppose that $(S,\cdot)$ is a magma such that $(ab)a = b$ for all $a,b \in S$. Then $a(ba) = b$ for all $a,b \in S$. I would imagine one can formulate many similar results with varying degrees of nontriviality.

@mechanodroid Do we have natural concrete examples of such magmas that are not groups to which this result applies?

There is a page on magmas on ProofWiki: proofwiki.org/wiki/Definition:Magma

Because you didn't have Lang. Just kidding but I think he did mention them near the beginning.

@metoo: In Lang's algebra, he does not mention magmas, I believe. He does however mention monoids.

There's something even more general than a proof: a sequence of words. And something even more general than a sequence of words is an array of pixels. Instead of studying proofs, why don't we study pixel arrays?

@user253751 exactly. :)