In economics, if utility is strictly increasing in t—the quantity of it consumed—then we would call it a type of a "good", and utility functions are often unbounded. What makes the ultimate choice of t finite is that utility from t is typically modeled as concave, while costs are convex. I think you might be able to find some literature on convex utility functions, but my impression is that there isn't much to study here:
If utility is strictly increasing in t (even accounting for cost of its inputs/opportunity cost etc.), then you will consume all of it that you can access. So if the stock of available t is finite, then you have reduced the problem to a simpler subproblem, where you allocate your resources (minus what you spent on t) to everything else. If t is infinite, then you have attained infinite utility and need not make any other decisions, so the problem is again uninteresting.
Of course, the assumption that utility is strictly increasing in t, regardless of circumstance, is a strong one, but I think it is what you're asking.