Actually, as opposed to what I claimed before, we don't need computational complexity bounds for this definition to make sense. This is because the Solomonoff prior is made of computable hypotheses but is uncomputable itself.

Given $g>0$, we define that "$\pi$ has (unbounded) goal-directed intelligence (at least) $g$" when there is a prior $\zeta$ and utility function $U$ s.t. for any policy ${\pi }^{\prime }$, if $E_{\zeta {\pi }^{\prime }}\left[U\right]\ge E_{\zeta \pi }\left[U\right]$ then $K\left({\pi }^{\prime }\right)\ge D_{KL}\left({\zeta }_0||\zeta \right)+K\left(U\right)+g$. Here, ${\zeta }_0$ is the Solomonoff prior and $K$ is Kolmogorov complexity. When $g=+\infty$ (i.e. no computable policy can match the expected utility of $\pi$; in particular, this implies $\pi$ is optimal since any policy can be approximated by a computable policy), we say that $\pi$ is "perfectly (unbounded) goal-directed".

Compare this notion to the Legg-Hutter intelligence measure. The LH measure depends on the choice of UTM in radical ways. In fact, for some UTMs, AIXI (which is the maximum of the LH measure) becomes computable or even really stupid. For example, it can always keep taking the same action because of the fear that taking any other action leads to an inescapable "hell" state. On the other hand, goal-directed intelligence differs only by $O\left(1\right)$ between UTMs, just like Kolmogorov complexity. A perfectly unbounded goal-directed policy has to be uncomputable, and the notion of which policies are such doesn't depend on the UTM at all.

I think that it's also possible to prove that intelligence is rare, in the sense that, for any computable stochastic policy, if we regard it as a probability measure over deterministic policies, then for any $ϵ>0$ there is $g$ s.t. the probability to get intelligence at least $g$ is smaller than $ϵ$.

Also interesting is that, for bounded goal-directed intelligence, increasing the prices can only decrease intelligence by $O\left(1\right)$, and a policy that is perfectly goal-directed w.r.t. lower prices is also such w.r.t. higher prices (I think). In particular, a perfectly unbounded goal-directed policy is perfectly goal-directed for any price vector. Informally speaking, an agent that is very smart relatively to a context with cheap computational resources is still very smart relatively to a context where they are expensive, which makes intuitive sense.

If we choose just one computational resource, we can speak of the minimal price for which a given policy is perfectly goal-directed, which is another way to measure intelligence with a more restricted domain. Curiously, our bounded Solomonoff-like prior has the shape of a Maxwell-Boltzmann distribution in which the prices are thermodynamic parameters. Perhaps we can regard the minimal price as the point of a phase transition.