I'm looking for research on the utility of campaign money, specifically in USA elections. Is there a point at which more money doesn't significantly increase a candidates chance of winning? How would we go about determining that?
There has been research on this. Certainly both parties look at the effects of spending and how they can leverage it for electoral advantage.
One problem though is that there are confounding effects. For example, Obama was more popular than Romney in 2012. Obama both outspent Romney and won the election. So is that a sign that spending more wins elections? Or is it an example of being more popular being good for both contributions and winning?
There is some published research. For example, here is a Freakonomics article with links to some papers.
Among other things, there are some natural experiments. For example, self-funded candidates have high spending even with low contributions. If the primary benefit of higher spending is direct, they should be more effective than other candidates. If popularity instead leads both to high contributions and to winning, they should be less effective than other candidates with similar spending. And the latter is what we actually observe. Self-funded candidates tend to only do well when they also raise significant funds from contributors. Some recent examples of failed self-funded candidates are Linda McMahon, Meg Whitman, and Carli Fiorina.
Generally, money is most effective in down ballot races, where information about the candidates is much rarer. Once the candidates become well known (ex. Jeb Bush), more money is mostly only effective at improving voter turnout for one's team, as opposed to convincing opponents.