Define a magma $\langle G,*\rangle$ with
$\forall a,b \in G,\quad a*b \in G $
$\exists 1(a*1=a\;\text{ and }\;1*a=1)$
$\forall a \in G, \exists b (a*b=1)$
$\exists c \forall a \in G, \quad(a*c=1)$
(associativity and commutativity does not need to always hold)
Could you please let me know the name of these kinds of magma?
Thanks for your comments. This looks like a kind of unital loop-like object.
Finally someone asked for an example. Well, I am surprised that nobody noticed that I am just restating the properties of one of the most seen operations.
Your structure is modeled off of exponentiation. But if its identity is not two-sided, it has an absorption element, and is not associative, it is not very "group-like." Indeed, despite monoids having absorption elements, their identities and absorption elements are still two-sided, so it's not even monoid-like. I will go on a whim and guess there is not a standard name for the structure you describe (or if there is, it is very obscure and the structure is something I'd call uninteresting).
Given how natural of an operation exponentiation is, you might be tempted to think it odd abstract algebra hasn't distilled from it a type of algebraic structure. But I think it's best instead to interpret it in a different context: right actions.
Let $R$ be a set and $(N,\circ)$ an algebraic structure (letters chosen reminiscent of $\mathbb{R}$ and $\mathbb{N}$). Consider an operation $\wedge: R\times N\to R$. Here is an axiom we might like it to satisfy:
$$ (r\wedge n_1)\wedge n_2=r\wedge (n_1\circ n_2) \tag{1}$$
This means that every element of $N$ acts as a function of $R$ "from the right." In practice, it'd make the most sense for $R$ and $N$ to be groups or at least monoids. One common example of this is right group actions. Natural occurrences of right group actions are of $G$ on $\hom(G,X)$ when $X$ is a set, also known as the contragredient action in representation theory, of which $S_n$ acting on a Cartesian power $X^n$ or a tensor power $V^{\otimes n}$ are special cases, and monodromy actions of fundamental groups on fibers of covering maps.
If $(R,\cdot)$ itself has an operation, then we might add another axiom:
$$ (r_1\cdot r_2)\wedge n=(r_1\wedge n)\cdot (r_2\wedge n) \tag{2}$$
This says that $(N,\circ)$ acts from the right by endomorphisms of $R$. (Or, if $N$ is a group, then it acts by automorphisms.) In general, to any monoid $(R,\cdot)$ there is an associated monoid $(N,\circ)$ of right endomorphisms. An example where this occurs is in Galois theory: Galois groups act by automorphisms (with respect to both addition and multiplication operations in fact), albeit from the left instead of the right (but any left action may be converted to a right action and vice-versa).
Another important example of where this occurs is conjugation in a group. Conjugation by a specific element is usually treated as a left automorphism - that is, $\varphi_g(x):=gxg^{-1}$ satisfies $\varphi_g(xy)=\varphi_g(x)\varphi_g(y)$ - but occasionally group theorists will prefer a compactified and suggestive exponential notation $x^g:=g^{-1}xg$ which is a right action by automorphisms. In this notation, both $(xy)^g=x^gy^g$ and $(x^g)^h=x^{gh}$ are true, exactly like exponentiation.
Finally, yet another axiom we could add could be
$$ (r\wedge n_1)\cdot (r\wedge n_2)=r\wedge (n_1\oplus n_2), \tag{3}$$
where here we write $\oplus$ for an operation in $N$. If $(N,\oplus,\circ)$ is a ring, all three axioms makes $R$ an $N$-module! In particular, this means the positive reals $R=(\mathbb{R}^{>0},\cdot)$ is a one-dimensional real vector space over the reals $N=(\mathbb{R},+,\cdot)$. This is an extremely important intuition to have when using all of the exponent laws in intermediate algebra (while of course an elementary algebra student will not have access to lingo like "vector space" or "action" or "homomorphism," this concept is still ultimately behind everything).