Did people ever live at the Tower of London?

It was built first as a fortress stronghold and finished in the year 1100. After the battle of Hastings in 1066, William the Conqueror needed to control the Londoners if he was going to have a quiet time as the new king (who they didn't want), so he built fortresses all around the old Roman wall.

Quote:
"After his coronation in Westminster Abbey on Christmas Day 1066, the new king withdrew to Barking in Essex, ‘while several strongholds were made ready in the City to safeguard against the fickleness of the huge and fierce population, for he saw that his first task was to bring the Londoners completely to heel’."

So the Tower has been fit for a king, and Kings and Queens have certainly stayed there, but its never been a primary or permanent residence.

Yes. Prisoners and Yeomen Warders. +4

Definitely. And they still do: there is a military Governor in residence, and the Yeomen Warders still live there. It has always been a functioning military base, and often a prison, with resident staff as well as prisoners.

The tower's primary function was a fortress, royal palace,
and prison - particularly for high status and royal prisoners,
many of whom didn't usually live there for very long!
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The tower was also used as a prison for German prisoners
of war throughout WW2.
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Rudolf Hess, Adolf Hitler's deputy, was imprisoned there for
4 days after he came to Britain seeking peace in 1942.
An RAF Wing Commander was placed in the same cell, pretending
to be a Luftwaffe officer, so he could spy on Hess.
http://en.wikipedia.org/wiki/Tower_of_London

Many. Some well known ones were the nephews of King Richard III, who were probably murdered there. Also, Mary, Queen of Scots, for many years. She was executed by Elizabeth I. Fear on Elizabeth's part.