Final answer:
The Puritan thinker who wrote 'The Pilgrim's Progress' while in prison in the 1670s was John Bunyan, a significant figure in religious literature.
Step-by-step explanation:
In the 1670s, the Puritan thinker who began writing 'The Pilgrim's Progress' while in prison for illegal preaching was John Bunyan. He was not only a preacher but also a prolific writer, and 'The Pilgrim's Progress' is considered one of his most significant and enduring works. It is an allegorical journey of a Christian, named Christian, navigating his way from the City of Destruction to the Celestial City, representing the journey from sin to salvation.