In the Doctor Who episode 'Listen', towards the end we see
the Doctor as a young child
We never see the actor's face, and he's not mentioned in the end credits nor on the Wikipedia page for the episode, but some young lad is going to be able to say throughout his acting career that he once
played Doctor Who!
Who is the actor?
According to Wikipedia:
A young version of the character appears in "Listen" (2014), portrayed by an uncredited child actor.
I agree with Richard who said it was likely a relative or family friend of the crew. However they found them, I'm sure if they didn't credit them, they would find it impolite for us to put their name.
As mentioned in another answer, the uncredited actor is a boy named Michael Jones. This was confirmed in Doctor Who Magazine #519 in a feature about William Hartnell.
The Tenth Planet in October 1966. Since Hartnell's death in April 1975, several actors have given us their First Doctor. Richard Hurndall in The Five Doctors (1983) was the first to step into Hartnell's shoes, metaphorically speaking; then came former companions William Russell and Peter Purves, in various Big Finish audios; voiceover artist John Guilor, albeit briefly, in The Day of the Doctor (2013); and Michael Jones played him as a child (probably) in Listen (2014).
The Doctor Who Daily Blog on AminoApps (usually a reliable source of information about the show) indicates that he was chosen for his resemblance to a young Hartnell.
Archive footage inserted in post-production indicated that this was the same barn occupied by the War Doctor in The Day of the Doctor. Michael Jones was cast as the young Doctor, after the crew noticed his resemblance to a photo of William Hartnell as a child.
No indication is given as to where they located him from and/or whether he was a relative of the crew or production team (as is common for uncredited child cameos on BBC programmes) or from a talent agency.
