By a generalized Euler characteristic $\chi$, I mean an isomorphism invariant $\chi(V)$ inside some abelian group $A$, defined for every varietiy $V$ over a field $k$, with the property that, for all varieties $X$ and open subvarieties $U$, we have $$\chi(X)=\chi(U)+\chi(X\setminus U).$$ (In other words $\chi$ is a group homomorphism $K_0(Var_k)\to A$ from the Grothendieck ring of varieties.)
There are many generalized Euler characteristics, for example the ordinary Euler characteristic (with compact support), the Hodge numbers (at least for smooth projective complex varieties), the number of rational points (over finite fields), etc.. However, all these examples are of motivic origin, i. e. they factorize over the map assigning to a smooth projective $X$ the class of its motive in the Grothendieck ring of Chow motives.
The only Euler characteristic I know of for which this is not the case is the map $K_0(Var_k)\to \Bbb Z[SB]$ where $SB$ denotes the monoid of stable birational equivalence classes, sending a smooth projective X to its equivalence class.
Are there any other known Euler characteristics factorizing neither through $K_0(Var_k)\to K_0(\mathcal Mot_k)$ nor $K_0(Var_k)\to \Bbb Z[SB]$?
(Here, of course, $\chi$ should be somewhat sensible, in the sense that the structure of target $A$ should not be to complicated (so as to exclude the universal Euler characteristic $id:K_0(Var_k)\to K_0(Var_k)$.)