This answer address both the original question and the poignant comment by YCor.

Such a concept has already been formalized, pun intended, by H. Schoutens around 2009-2012 in two preprints, found found here: https://arxiv.org/pdf/0910.0684 and https://arxiv.org/pdf/1004.4260

One of the main innovative ideas here is to use the functor of points  perspective of Algebraic Geometry to pick up a non-empty compliment $X^{\circ} \setminus X_{red}^{\circ}$ where $Y^{\circ}$ denotes the functor of points from finitely generated $k$-algebras to sets given by a separated scheme of finite type over a field $Y$.  

One can define several interesting *motivic sites* on the category of separated schemes of finite type over a field as different natural collections of *sieves* which happen to form a distributive lattices (so that scissor relations hold and fiber products work as multiplication). 

The most interesting of these sites is the formal site $\mathbf{Form}_k$ whose Grothendieck ring I will denote by $\bf{H}$. It is the right concept in my view as it takes into account thickenings of arbitrary order. Of note, in $\bf{H}$, 

$$[\mathbb{P}_k^1] = \mathbb{L}+ \widehat{\mathbb{L}} $$

where $\widehat{\mathbb{L}}$ denotes the class of the formal sieve given by the functor of points of the completed affine line at the origin. 

Note further that there is a natural surjective ring homomorphism 
$\bf{H} \to K_0(\bf{Var}_k)$ given by sending the class of a formal sieve $[\mathcal{X}]$ to $[\mathcal{X}(k)]$ and if you complete these rings along the dimensional filtration, you obtain continuous ring homomorphisms of topological rings (cf., Theorem 2.1 and Lemma 2.4 of my paper here: https://doi.org/10.1016/j.jsc.2016.08.011)

Note then that it is immediate that $[X^{\circ}]$ is sent to $[X_{red}]$. It is slightly less immediate but true that $[(\widehat{X}_Y)^{\circ}]$ is sent to $[Y_{red}]$ so that in particular $\widehat{\mathbb{L}} \mapsto 1$ under this ring homomorphism. 

A few things I find interesting about this ring: 

1) You get to localize by a bunch of stuff for free since many things get mapped to 1.  
2) Relatedly, it contains a highly nontrivial subring $\textbf{H}_0$ formed by all zero dimensional formal motifs which gets mapped to $\mathbb{Z} \subset K_0(\mathbf{Var}_k)$. 
3) *It is very rigid*: I suspect that  $[X^{\circ}] = \mathbb{L}^n \implies X \cong \mathbb{A}_k^n$. 
4) The possibility of some type of rationality of the motivic generating series of the form: $P_m : = \sum_{n=0}^{\infty} [(\underline{Hom} (D_n, D_m))^{\circ}]\mathbb{L}^{-r_n}t^n$ where $D_i : = \text{spec}(k[t]/t^{i+1})$ and $r_n = n - \lceil \frac{n}{m}\rceil+1$ such that $P_m \mapsto (1-t)^{-1}$. 
5) Relatedly, the possible rationality of $A_n := \sum_{n=0}^{\infty} [(\underline{Hom} (D_n, D_n))^{\circ}]\mathbb{L}^{-n}t^n$.