A composite event occurs when all or many of a doll's imprinted personalities and skills reassert themselves in his or her brain. Such an event was considered theoretically impossible, since each imprint is wiped completely from the doll's brain after each engagement. However this event has occurred with Alpha due to an accident with the imprinting chair.
After a composite event it is hard to tell who is the dominant personality. Also, as evident with Alpha, a doll who suffers a composite event is not only aggressive but very intelligent as well. Alpha has proven this on multiple occasions, with the most notable incident being when he hacked the Dollhouse secure network and performed a remote wipe. This is impressive because not only the Dollhouse has a great amount of security, but because remote wiping was supposed to be impossible. This would mean that a composite doll can access the combined knowledge of their imprints' or the imprints are directly part of their personality.
Occurrences
The first known composite event occurred in 2008, when an accident in Topher Brink's Imprint Room caused at least 40 personalities to imprint at once. This caused Alpha to go insane while at the same time becoming exceptionally killed in combat, technology and survival. This combination made Alpha extremely dangerous, to the point where he was able to kill a Handler in a period of eight seconds, when the injuries he inflicted should have taken hours. (S1: "The Target", "Briar Rose", "Omega"; S2: "A Love Supreme", "Epitaph Two: Return")
Echo's composite event occurred in 2009, when Alpha kidnapped her, and imprinted at least 36 different personalities, except for Caroline Farrell, into her. This started a chain reaction that eventually allowed Echo to resist imprinting and wiping if she so disired, and allowed her to switch between retained personalities at will. Unlike Alpha, Echo did not lose sanity, but suffered severe headaches that gradually became worst each time a new personality or skill set was imprinted. (S1: "Omega"; S2: "Vows", "Meet Jane Doe", "A Love Supreme", "Stop-Loss")
Who has experienced a composite event
Alpha
After attacking Whiskey, Alpha had all his previous personalities imprinted into him due to an accident in the Imprinting Room, causing a composite event. Alpha, as a result has become a highly aggressive and intelligent being who is nearly impossible to trace. After the event, his first act was to destroy his original personality for 'abandoning' him. His skills are many and he seems hellbent on taking the Dollhouse down
Echo
After her composite event, Echo learned to imprint and wipe herself, and could change from one personality to another, all at will.
In Omega, Alpha abducts Echo and engineers a composite event in her, similar to his own. However, Echo does not show the psychotic tendencies exhibited by Alpha (which were probably caused by remains of his original personality), and refuses to kill Caroline (imprinted into an innocent bystander). She eventually returns to the Dollhouse and is wiped, apparently returning to her former status. At the end of the episode she whispers the word "Caroline" to herself, which can be either another sign of her growing self-awareness (which was happening long before the composite event) or an indication the wipe is not completely successful after a composite event.
Echo has shown signs of a growing self-awareness and personality, though whether these are symptoms of a natural composite event or her own evolution remain to be seen. In The Target, exposure to drugged water (possibly provided by Alpha) caused Echo to see past imprints meaning she could very well be close to a composite event and the right stimulants are the only thing now needed to trigger it. Interestingly, during the events of Briar Rose Echo seemed to show signs of composting when she remembered bits of her imprints and acted accordingly, fighting Ballard in self defense when she remembered their battle before, and recognizing Boyd as a friend when the imprint bond should have been wiped.
In recent episodes Echo is showing more and more retention of her previous imprints and in Epitaph One remarks to Paul Ballard (who became her handler early in the second season) that all of her imprints are active in her mind, and that this causes her to have extremely painful headaches. Echo's multiple imprints in Epitaph One and her composting in Omega show that her concurrent imprints are more cooperative with one another than Alpha's, or that Echo herself is better able to control them.
According to Boyd, Caroline Farrell aka Echo was born with a specific gene which allows her to resist the wipes and evolve in her Tabula Rasa state.