While (or whilst) musing in another thread about having changed my car registration from Pennsylvania to Texas, for some reason I began thinking about this question. I once had an Audi that I ended up registering in five different states during the ten years I owned it. In sequence: Colorado, California, Texas, Wyoming and Pennsylvania. I still have all the plates stashed away somewhere, and they were all fairly cool-looking.
So, can anyone beat that?
Georgia, Tennessee, Virginia, Ohio and Mississippi. That is only a tie, but I thought I’d mention it.
Florida, Illinois, Nevada & California.
(I made the mistake of telling the Navy I wanted to travel).
Tennessee, Mississippi, Georgia and back to Tennessee - only four. But if you look at the various TN counties (Hamilton, Rutherford, Davidson and Knox)…it adds up.
In order, CT, VT, RI, ME, CT, and now California. I win if you let me count CT twice…
One 1984 Toyota Corolla.
Arizona, Texas, Panama (the country), skipped Oklahoma, New Jersey and finally Texas again where it was sold.
Nothing like driving around in Oklahoma with expired Panamanian plates! 
Hey pilot141 did you drive it down to Panama? Back? or both?
kniz, no it was on a boat. A buddy of mine drove his Jeep Cherokee back - took off the catalytic converter, stocked up with cash and cigarettes (for “easing” discussions with any local police forces) and made it back in 10 days.
Five also: Kentucky (twice), California, Washington, Alaska, Tennessee.
Sir Rhosis
Another one with 5 states: WY, TX, NE, OR, SD
Maryland, Florida, Delaware and sold to BIL - taking it back to Maryland.