Unseen Stories (6)

The Fifth Doctor
As we've already seen, once John Nathan-Turner took over as producer, there was a conscious effort to cut back on the humour in the programme. One of these humorous elements had been the propensity of the Doctor to name-drop, usually in a funny way (the Doctor writing Hamlet due to Shakespeare hurting his wrist writing sonnets, dropping apples on Isaac Newton's head etc.).
As such, from The Leisure Hive onward we get fewer references to unseen stories.
As the new Doctor undergoes his regeneration crisis in Castrovalva, he mentally regresses back through earlier incarnations, and at one point recalls an incident involving the Ice Warriors, during which the Brigadier was present. Now, this might all be part of his mental trauma, jumbling up different events, but it might equally well be a genuine memory of an unseen adventure involving the Ice Warriors and UNIT.
In Four to Doomsday he states that he knew Sir Francis Drake (c.1540 - 1596), hero of the Spanish Armada rout. He also claims to have taken five wickets for New South Wales, but doesn't specify the date. The club was founded in  1856, but the match he played in must have been sometime after 1933, when the "chinaman" style of spin bowling is said to have been born. The term is no longer used, being now deemed racially offensive - Wisden, the cricketing bible, dropping it as of 2018. As a time traveller, the Doctor ought to have known that.
Of course, the First Doctor didn't even recognise cricket, but he knows all about it from his Fourth incarnation onward. Presumably all that time spent in exile on Earth during his Third incarnation is when he picked up a taste for the game.
The Doctor seems to know a lot of detail about the Terileptils, recognising the injuries of their leader as having come from the tinclavic mines on Raaga (not the sort of thing a text book would have told him).
Arc of Infinity seems to suggest that there has been an unseen return to Gallifrey at some point, as he seems very familiar with the current make-up of the High Council. He asks Damon how Leela is, though Damon was absent from The Invasion of Time. Some history with the new Castellan is also implied.
The King's Demons gives us the first mention in the series of the Eye of Orion, which the Doctor seems to have visited on more than one occasion. Unlike most other holiday destinations mentioned in the past, we'll actually get to see it in the very next story - The Five Doctors. This story throws up more potential "Season 6b" implications, with the Second Doctor recalling that the Time Lords wiped the memories of Jamie and Zoe. He couldn't possibly know this before going to the Brigadier's UNIT reunion party, and he's travelling alone. (He's actually wrong about them remembering him, as the Time Lords allowed them each to retain their memory of their first encounter with him. It's the fact that Jamie knows who the Brigadier is that's significant). The Second Doctor also mentions an encounter with the Terrible Zodin, when the Brigadier definitely wasn't present.
The Third Doctor appears to recognise a Raston Warrior Robot, suggesting a previous encounter.
Then there's that trip the Fourth Doctor and Romana took to Cambridge...
It is suggested even more clearly that the Doctor and the Castellan have history. Chancellor Flavia then states that she has the honour of offering the Presidency of the High Council to the Doctor "yet again" - even though we've never seen her do this before.
Warriors of the Deep throws up all sorts of continuity problems regarding the Silurians and the Sea Devils. In some ways it is trying to say there were just the two Third Doctor encounters, yet the detail says there has to have been an unseen adventure involving the specific members of the Silurian Triad. He recognises a Silurian battle cruiser, and the Myrka, though neither ever featured before.
There is an unseen trip to 1643 to drop off the temporally displaced Will Chandler after the events of The Awakening.
The Doctor seems to have learned about developments in the Dalek-Movellan war, knowing all about the Movellan gas. Note how the Doctor knows exactly where and when to go to visit this conflict, in its active stages, in The Pilot.
Finally, for the Fifth Doctor, we have him visiting Androzani Minor, which he says he has been to before. He seems to have known Professor Jackij, at a time before he discovered the cure for spectrox toxaemia, as he hasn't heard of spectrox before.