Since specific pre-determined axioms might have problems, to formalize the problem to give a reasonable value to integrals $\int_0^\infty f$ (with real $f$) where each $F(x)=\int_0^x f$ reasonably exists for each finite $x$, I suggest to look at the universal compactification of $[0,\infty)$ taking into account the amenability of this additive semigroup (in the same way as amenability of the additive semigroup of natural integers gives Banach limits of bounded sequences). [The standard reference for rings of continuous functions is Gillman - Jerison; I will not give references for amenability, only note Wagon's book that relates in detail failure of amenability to existence of paradoxical decompositions like Banach - Tarski - Hausdorff]

The above functions $x\mapsto F(x)$ are continuous, hence when bounded (oscillating integrals) have a unique real continuous extension to the universal compactification $\beta[0,\infty)$; as value at infinity the "universal value" is then an element, call it $F(\infty)$, of the algebra $C(X)$ of continuous real valued functions on the remainder $X=\beta[0,\infty)\setminus[0,\infty)$. $F(\infty)$ has a unique real value, i.e. $F$ extends by continuity to the one point compactification of $[0,\infty)$, iff it is a constant function in $C(X)$. Commutativity, hence amenability, of $[0,\infty)$ gives (independence of $F(x)$ from the value of $f$ in fixed finite segments $[0,x]$ being automatic) that we can also chose a "almost reasonable" single real value instead of a family $F(\infty)$ of such values, by chosing a suitable positive measure $\mu$ on the remainder $X$ and integrating: $\int_X F(\infty)d\mu$ (choosing a Dirac's delta for $\mu$ means chosing one of the points of the remanider $X$ i.e. choosing a ultrafilter of co-zero sets in the original space).

This is perhaps the best that can be done for associating reasonable values to boundedly oscillating integrals. When $F$ is not bounded, it is still continuous; by Gelfand - Kolmogorov, the maximal spectrum of the ring $R$ of (possibly unbounded) real continuous functions is the same (universal compactification $\beta$) as that of the ring of bounded real continuous functions, and $F(\infty)$ is then again definable as function on the remainder $X$, but this time its values in each point at infinity (i.e. maximal ideal $m$ of the ring $R$) need not real numbers, but elements of a noarchimedean real-closed extension field $R/m$ of the real numbers (that field depends upon $m$, but a common extension exists, by the amalgamation property of real closed fields: Hodges, model theory, pp. 384 - 386). So one has reached nonstandard analysis and "orders of infinity" like in the previous answers & comments. What changes for general $F$ when comparing with the subcase of bounded $F$ is that this time no obvious averaging process exists to obtain a single hyper-real number instead of a family $F(\infty)$ of such numbers. [Or, more exactly, I know no theory of measure and integration with hyper-real valued functions and measure that would easily integrate all $F(\infty)$ in the same way as the standard theory works for the real, archimedean, case]

The preceding considerations are by construction translation invariant, but the action of the group of dilations (last axiom) is not considered. The group generated by translations and dilations (i.e. the group of affine transformations of the real line) is the semidirect product of two abelian groups, hence soluble hence amenable. So perhaps it might be possible to obtain a kind of suitable invariance / covariance also for dilations, despite the fact that the explicitly noted axioms have problems.