Catholic theologian Karl Rahner hypothesized that reincarnation might be one way that Purgatory operates, especially for those people who (on earth) did not live in a circumstance that allowed them to freely choose God (perhaps because they were not exposed to the gospel, or perhaps because they had to struggle all their lives merely to survive war, hunger, or various types of oppression).
Catholic Brother Steindl-Rast agrees:
"From my own personal views, I have studied reincarnation sufficiently that, when somebody asks me, "Do you believe in reincarnation?"--which often happens after a lecture because it is a very popular topic--I simply say, "Yes, I do because I am a Roman Catholic and our name for it is purgatory, but we haven't developed it."
Wait a minute... in all my roman catholic education I have
never heard that reincarnation is part of our beliefs
Purgatory is part of the Catholic beliefs, but you have to consider that there are HUGE differences between catholic theology and catholic folklore. Catholic theology invented/discovered purgatory because it seemed theologically impossible that someone good enough to merit heaven but not "perfectly good" would be immediately allowed to see God, or perhaps he could not withstand such meeting, hence the need for a
temporary state of purging from all the remnants of evilness (or non-goodness) before going to heaven.
It is catholic folklore of the common people (who as usual needs to fantasize and visualize everything, but cannot do so without remaining tethered to our life-based concepts of time, space and human senses) which started to imagine purgatory as an actual, physical place where to spend years in "mild" suffering.
I suppose you can take the latter view and stretch it to mean that such place is on earth, and such time is measured in additional lives, until someone is finally enlightened. This is actually not really different from Buddhist concept of reincarnation, but as far as I know it is completely rejected by the Catholic Church.
I have actually thought in the distant past something quite similar, except that I was wondering if hell might mean reincarnation. The evangelical references to evil people been condemned to live in the flames of Gehenna forever + the fact that "Gehenna" was actually the name of a very hot region in the middle east, these made me wonder if Jesus could have meant that evil people are condemned to live and suffer on earth again and again. "Forever" meaning "forever as they are evil", but if finally enlightened they would be evil no more and that "forever" would not apply to them anymore. This was just my own mind-rambling as a teenager tho...
It depends on how sadistic is your view of hell and how bucholic your view of purgatory. Some people lives are so unfortunate that hell could be very much like that, but I suppose most people think of hell as continuous physical torture which is unlikely in real life (thought not so infrequently heard of).