I am retired and live in three states, but never more than 5 months in any of them, to whom do I owe state taxes

Assuming that you have no earned income (salary or wages) - then it's generally the one in which you spend the most time. But perhaps more importantly - in which state do you vote, and what driver's license do you have?

What about my income from pensions and tax deferred annuities? I thought you had to pay taxes in a state where you live at least 6 months plus one day. If this is so, am I exempt from taxes in any state?

No, you're not exempt - you owe taxes to the state of your principal residence (your "domicile"). Everybody has one and only one domicile (although you may live in multiple states) - you need to determine which of the three is yours. There is no hard and fast rule, but in general it's the place that you consider your permanent home, and where you spoend more time (even if less than 1/2 year) It is most likely the place where you have your drivers license and are registered to vote. If you own a home in one state, and rent in the other two, that may be evidence of your domicile. Be aware that if in the past your domicile was in one of these 3 states, the onus is on you to provide evidence that you have intentionally relocated your domicile to a different state.