The answer to this depends to a large degree on a person's political ideology. Conservatives tend to argue that most public functions could benefit from competition while liberals take the other position. For example, conservatives tend to be strongly in favor of charter schools while liberals prefer that regular public schools should not have to compete for funding.
Another public function that might benefit from more competition is prisons. States spend huge amounts of money on prisons. It is possible that allowing private companies to compete to run prisons would result in lower costs for the government.
Finally, conservatives would argue that health care is an area where more competition is needed. They would argue that changing laws to make it possible for more insurance companies to compete for customers would lead to greater efficiency. Liberals, by contrast, often argue for a single-payer, government program for health care.
Thus, education, prisons, and health care are functions that, to some people, would benefit from greater competition.