What was erased from Jack Harkness's memory?

Question

I just watched The Doctor Dances (Doctor Who, S01E10). Captain Jack Harkness said the Time Agency took two years worth of memories from him.

[Jack's spaceship]

ROSE: So, you used to be a Time Agent now you're trying to con them?
JACK: If it makes me sound any better, it's not for the money.
ROSE: For what?
JACK: Woke up one day when I was still working for them, found they'd stolen two years of my memories. I'd like them back.
ROSE: They stole your memories?
JACK: Two years of my life. No idea what I did. Your friend over there doesn't trust me, and for all I know he's right not to. Okay, we're good to go. Crash site?

The issue was never brought up again in either Doctor Who or Torchwood.

So what was erased from his memory?

Answer 1

You are correct in saying that The issue was never brought up again in either Doctor Who or Torchwood.

Jack's missing memories were somewhat addressed in Big Finish Productions' audio play series: The Lives of Captain Jack (2017), "Month 25" (S01E04), released June 6, 2017, in which a young Jack's time at the Time Agency is explored, and we uncover some of the mystery behind the "2 years of missing memory".

He’s the young star of the Time Agency, and his whole life is about to fall apart. But that’s not going to stop him winning.
_{—Official synopsis, Big Finish Productions}

During this episode, we learn the following things:

• Jack Harkness' real name is Javic Piotr Thane
• He was suspended from the Time Agency with docked pay for visiting 19th Century brothels

But not much else is explored about these mysterious two years (much to the delight of fans of his mystique, and much to the chagrin of fans who need to find out more). Indeed, it's during this episode that Jack finds out he's lost two years of his memory, and we get his immediate reaction, voiced by John Barrowman.