More recordings like "The Trinity Sessions"?

It's a very different style of music, but Todd Rundgren was pretty well known for employing this approach from time to time - with varying degrees of success. I couldn't tell you every release that he took that way, but there are several that I'm aware of. The most successful example (IMHO) of those TR records is Nearly Human, the least successful is probably Second Wind (IMO the record, ahem, blows).

Marty

A very different kind of music but "A Meeting By The River" by Ry Cooder and Vishwa Mohan Bhatt was recorded an hour after the two first met and reportedly was almost completely improvised. A remarkable recording.

Joe Jackson - "Big World"

Thanks! Will check them all out.
Steve

Elvis Costello's album My Aim is True was recorded for around 38,000 dollars and the rumour is the first take was what was pressed to vinyl.