tl;dr: Short answer is that cardinal utility of rational person is in large part of a literature is derived from the von Neumann and Morgenstern expected cardinal utility framework. In such framework utility must be bounded by our definition of what rationality is. So in such framework the short answer is simply that utility has to be bounded for person to be considered rational.
There are also expected utility frameworks that allow expected utility to be unbounded (see review by Fishburn, 1976). However, these are not very widely used as they can often lead to paradoxes and do not seem to offer any special insight. I suppose this is due to the strong influence of instrumentalism on economic thinking. A researcher would prefer a utility framework that is able to provide some testable predictions rather one that just results in a paradox and hence offers no useful testable prediction. So even if someone could perhaps consider concept of unbounded utility more elegant its instrumental value in doing research would be close to none (if in particular case it leads to unresolvable paradox).
Full Answer:
The utility needs to be bounded in order to avoid paradoxes such as the St. Petersburg paradox (for more nuanced overview see this entry in Stanford encyclopedia of philosophy). As a matter of fact a rational person's utility should be bounded as suggested by Arrow (1970) precisely in reference to the paradox above.
More general actually cardinal utility will be bounded by the axioms that were used to derive the expected cardinal utility in the first place. Following Neumann and Morgenstern (1947) Theory of Games and Economic Behavior, an expected utility from gamble can be described by the so called von Neuman-Morgenstern equation:
$$E[u(g_i)] = \sum_j u(X_{ij})p_{ij}$$
where $u$ is utility $g_i$ is gamble $X$ is an outcome and $p$ is a probability. Furthermore, for the above to be utility we must have some continuum of gambles for which:
$$g_i,g_j \in \mathbf{G}: g_j \succeq g_i \implies E[u(g_j)]\geq E[u(g_i)] $$
Now given this we can ask (as the author of the paper did) what would be properties of such utility function?
Now it turns out that basic rationality constraint where preferences satisfy transitivity, completeness, continuity and independence imply that the utility has to be bounded.
The completeness axiom states:
$$ \forall x,y \in \mathbf{X}: x \succeq y \vee y\succeq x \vee y \thicksim x $$
that is we can order all our options in terms of preference in some way.
The transitivity axiom states:
$$ \forall x,y,z \in \mathbf{X}, \text{ if } x \succeq y \wedge y \succeq z \implies x \succeq z$$
So if someone likes $x$ more than $y$ and $y$ more than $z$ then $x$ must be preferable to $z$
The continuity $z \succeq y \succeq x$, then there must be some probability $p$ such:
$${px,(1-p)z} \thicksim y $$
This implies that no outcome $x$ is so terrible that you would not take up some gamble involving $x$.
by independence axiom if $y \succeq x$ then for $z$ and some probability $p$
$${px,(1-p)z} \preceq {py,(1-p)z} $$
This axiom states that if two outcomes have the same probability, we should evaluate the two alternatives independently of what we think that the outcome is.
The above axioms are how we define rationality in expected cardinal utility. There are of course different possible specifications of utility functions but most modern research relies on the von Neuman-Morgenstern type (or related utility functions).
Now these rationality requirements - which are axioms so they are by definition what rationality is in this context, simply demand that no outcome can produce infinite utility. In order to see this we can try to do proof by contradiction - suppose that there is a gamble wehere: $x = 1€$, $y=100€$ and $z= \infty €$ and that $u(X)=x$. In this case clearly $z \succ y \succ x$ but there is no $p$ for which the continuity axiom holds. Since the continuity axiom would be violated our basic axioms of what rationality is would not hold and a person with unbounded utility would cease to be rational (within the context of von Neumann and Morgenstern framework).
