Utility is a function of the person, the environment, and the thing. There is no 'utility of a dollar'. There is only 'utility of a dollar to me in this circumstance'.
So if one idolizes wealth then clearly they have assigned high utility to money in a non-diminishing returns sense. It is then rational for them to match their utility curve. Life is full of threshold systems. Given that, it would be very unusual for utility curves to not be deformed by them.
It's easy to construct a few scenarios with locally convex curves:
* You're in debt to the mafia and you will die if you don't pay them your $100k debt. You have $10k dollars. Do you spend the $10k to try a long shot of making $100k?
* You're poor and make $15/hr. If you save you will have an extra $1000 a year in savings. You will never own a home. You will never have time to study for a better job. The magic of compound interest with contributions is in the regularity, not so much in the amount. Put $1000/yr in a calculator and $998/yr in a calculator (assume it compounds 3% every year) and see the difference you get over 60 years. It is insignificant on a 60 year horizon. $2 will buy you a powerball ticket, and maybe out of this life.
* You're an immigrant. You need a few million dollars for your investor visa. Your current visa runs out next year and then you return to a slum.