The two major examples that come to mind are Bell Labs (run by AT&T) and IBM (in the 1950s - 80s era).

Assuming you are referring to IBM's mainframe business, they did not really have a monopoly; they were just a dominant player. Competitors at that time included Amdahl, Burroughs, UNIVAC, NCR, Control Data, Honeywell, General Electric and RCA. Amdahl even offered products that were compatible with IBM's mainframe offerings and could run software developed on/for IBM.

You seem to interpret "progress" as "lower prices" :-)

I see how it can look like that. But no, they are two related but seperate effects of a healthily competitive market.

...a potential monopoly is the carrot that drives a lot of innovation

That is a statement that I can agree with. (And I wouldn't count a severely dominant player as a monopoly; a dominant player can drive a lot of progress as the smaller fish frantically try to find a way to one-up it and pull in customers)

While I think that Clarity is misunderstanding Peter Thiel (Thiel says that a potential monopoly is the carrot that drives a lot of innovation; Clarity wrongly interprets this as "monopolies drive progress")

I read Thiel to be making the second argument as well, because monopolies have capacity to make the profits necessary for major R&D spending, as well as the capacity to benefit directly from major R&D spending. No individual commodity producer making zero economic profit has an incentive to invest in better methods of producing their commodity, as they correctly believe that the innovation will rapidly spread and benefit the consumers, rather than them.

The short way to visualize it is that the market for innovations has producer and consumer surplus, and only monopolists can capture both at once, which would suggest a monopolistic industry would spend more on innovation than a competitive industry. (This is true for some innovations and not others; monopolies should be expected to be better at developing basic science related to an industry, and competitions should be expected to be better at determining customer preferences.)