Fix a field $k$. The Grothendieck ring $K_0(\mathrm{Var}_k)$ of varieties over $k$ is defined as the quotient of the free abelian group on isomorphism classes of algebraic varieties by the scissor relation. The notation suggests that there might be higher $K$groups $K_i(\mathrm{Var}_k)$ as well, but naive attempt at defining such an object fails as $K_0(\mathrm{Var})$ is not defined as $K_0$ of an exact additive category. Is there a reasonable definition of these groups nonetheless?

Torsten Ekedahl proposed a definition of higher Grothendieck groups of varieties. Unfortunately it seems that he never wrote anything down on this topic before passing away. Torsten had quite a large number of unfinished mathematical manuscripts and projects. I don't know what happened to them, although surely someone at Stockholm University has taken care of them. I saw him give a talk about higher Grothendieck groups of varieties at Gerard van der Geer's birthday conference on Schiermonnikoog in 2010. The abstract is available online: http://wwwirm.mathematik.huberlin.de/~ortega/schierm/
Edit: I found my (sketchy) notes from the 2010 talk!!! Any mistakes in what follows are my own; I know close to nothing about Ktheory today and I knew literally nothing in 2010. Anyone who is more knowledgeable than me is very welcome to edit the following: He begins by recalling the definition of $\newcommand{\Var}{\mathbf{Var}}K_0(\Var_S)$ for a base scheme $S$. There is nothing new here. Then he declares his intention to define higher Kgroups. He makes a list of properties that such higher Kgroups should have (I might not have written down all of them):
So far we could just set the higher Kgroups to be zero. We want a nontriviality condition.
He goes on to discuss generally how to define algebraic Ktheory. You want a suitable category, such that the homotopy groups of its nerve are the Ktheory groups. He mentions Quillen's Q construction but says that he will follow Waldhausen's approach. Waldhausen's idea is to associate a "simplicial category" $\newcommand{\C}{\mathscr C}s\C$ to a category $\C$. He notes that there is a subtlety here, in that the simplicial identities $d_i d_{j} = d_{j1}d_i$ etc. need to be strict. For $\Var_k$ he defines $(s \C)_n$ to be the category with objects $$ \varnothing \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X_n $$ where all injections are closed immersions of $k$varieties, morphisms are isomorphisms of such diagrams. All the $d_i$ are "what you expect" except for $d_0$, which maps to $$ \varnothing \hookrightarrow X_2 \setminus X_1 \hookrightarrow X_3 \setminus X_1 \hookrightarrow \cdots \hookrightarrow X_n \setminus X_1. $$For each $n$, $N((s\C)_n)$ is a simplicial set. $N(s\C)$ is a bisimplicial set, so more or less a simplicial set. We define $$ K_i(\Var_k) = \pi_{i+1} N(s\Var_k).$$ He goes on to discuss Waldhausen's additivity theorem. Consider the category of pairs $X \hookrightarrow Y$ of closed immersions. There are three functors to $\Var_k$ mapping to $X, Y$ and $Y \setminus X$ respectively. These give three functors $K_i(\Var \hookrightarrow \Var) \to K_i(\Var)$ and the additivity theorem says that two of these sum to the third. Claim: He can prove this theorem for his definition of Kgroups. He notes that all his constructions mirror those of Waldhausen for topological spaces. The biggest difference is that Waldhausen's uses the existence of a quotient $Y/X$ for $X \hookrightarrow Y$. In particular one needs to give a different proof of the additivity theorem but this is possible. My notes end here. Of course the first question one asks is whether this definition agrees with the one due to Inna Zakharevich, that Clark Barwick linked to in a comment. 

