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Similar to Has the Doctor ever observed himself, without revealing who he was? but with a slight difference.
Has the Doctor ever worked with himself to solve a problem where both incarnations are the same?
There are numerous examples ("The Three Doctors", "The Five Doctors", etc) of different incarnations working together - but has he ever worked with himself?
The 3rd Doctor and Jo Grant interact with future versions of themselves in Episode 1 of Day of the Daleks.
(The Doctor goes underneath the console. A moment later the doors to the lab open and a second Doctor and Jo are standing there.)
DOCTOR 2: Yes, of course, I remember now. Look, don't worry, my dear. I know you're alarmed but you needn't be.
DOCTOR: Yes, well I think that should do it. Why on Earth I never realised that.
(Then the Doctor sees himself standing there.)
DOCTOR: Oh, no. What are you doing here?
DOCTOR 2: Well, I'm not here. Don't worry. Well, that is, in a sense I am here, but you are not there. Yes, well, it's a bit difficult to explain really.
DOCTOR: This won't do at all. We can't have two of us running about.
DOCTOR 2: Yes, well don't worry. It will all sort itself -
(Flash, bang from the console and the duplicates vanish.)
JO: Doctor? What happened?
DOCTOR: Well, it's a very complicated thing, time, Jo. Once you've begun tampering with it, the oddest things start happening.
JO: But there was another you and another me. Well, where've they gone?
DOCTOR: Back into their own time stream, of course. Or do I mean forward?-- transcript
After the Doctor and Clara have finally made their way all the way to the heart of the TARDIS, the Doctor pushes through a glowing crack in the wall to meet his earlier self from right back at the start of the episode. From the transcript (emphasis mine):
DOCTOR: Oh, yes. Big friendly button.
CLARA: You're lying.
DOCTOR: Yep.
CLARA: To stop me freaking out?
DOCTOR: Is it working?
CLARA: Not so much.
(The future Doctor pushes through the crack.)
FUTURE DOCTOR: Doctor, Doctor. I'm from your future. We haven't got long to reset time.
(He vanishes. The grenade rolls across the floor to Clara. She picks it up.)
As a bonus, there are also at least two more occasions in the show where Matt Smith has interacted with himself on-screen, although in-universe only one of these selves is the Doctor, the other being:
in one case ("The Rebel Flesh"/"The Almost People"), a Ganger in the form of the Eleventh Doctor:
in the other ("Nightmare in Silver"), a Cyber Planner taking over the Doctor's body and appearing inside his head as a Borgified version of him:
In the BBC Children in Need special Space / Time, the 11th Doctor speaks to a version of himself from mere moments into his future.
DOCTOR: I'm setting up a controlled temporal implosion. It's the only way to reset the TARDIS but unless I find exactly the right lever to control the implosion we're all gonna die.
AMY: You don't know which lever?
DOCTOR: No, but I'm about to find out.
(A second Doctor steps into the TARDIS through the main doors)
DOCTOR 2: The wibbley lever!
DOCTOR: The wibbley lever. Thank you!
(The original Doctor throws the wibbley lever and steps into the TARDIS parked in the control room which then dematerialises)
You can watch both parts legally on YouTube, at least here in the UK.
The 11th Doctor meets himself in the episode The Big Bang
From the plot synopsis
The injured body of a future version of the Doctor appears and whispers something to his earlier self.
The whole plot of this episode can be seen in this awesome music video by Chameleon Circuit The Big Bang 2
Yes, this has happened a few times.. most notably in the Mini-Series Night and The Doctor
In episode 4 (Last Night), Doctor Who interacts with a second version of himself, and 3 different versions of River. They dont solve a problem exactly, but it's the longest interaction between multiple Doctor Who's of the same incarnation I can remember.
And yes, Calling him "Doctor Who" is a joke. And yes, calling it a joke makes it less funn.. never mind.
Not a direct interaction, but in Smith and Jones, the Doctor of the near relative-future sets up Martha's interactions with the Doctor of the relative-present.
Arguably, Blink is along those lines as well, as it's the Doctor's discovery of the future transcript that causes him to record the video that eventually gets transcribed in the first place.
If you were to stretch a point, it could be argued that in Logopolis, The Watcher (or Doctor 4.5) helped both his current self (4) and his future self (5).
As Nyssa says: "The Watcher. He was the Doctor all the time."
But like I said, it's probably stretching a point.
