One of my players, a monk, decided to ready an action as an earth elemental was about to attack. The trigger was the elemental moving within five feet of him. When this occurred he took his attack action, and then proceeded to take the extra attack as well. I was unsure at the time whether or not the extra attack would also happen and I haven't had much luck figuring it out.
The 'Extra Attack' feature is worded so you only benefit from it when you attack in your turn.
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn
Reacting to something is not in your turn, it's in the turn of the triggering creature.
Extra attack, page 79 for monks says (emphasis mine)
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
The Ready action lets you take the Attack action on someone else's turn, and thus extra attack does not apply.
I would say, strictly by the rules...it's a shady area. The Monk in question has opted to forgo acting on this round to ready an action on the onset of a trigger. Technically speaking, this would then be their action, but is it their turn? This is really the question. Keep in mind, 5e DnD does not have the same concept of turns as previous versions.
If you look at PHB 189, it says that a round is about 6 seconds, and each participate in battle takes a turn in that round. Since the player has opted to ready an action, but not take their turn, then when the trigger occurs, it becomes their turn. Otherwise, if the trigger does not happen during that round, they forfeit their turn.
So the crux comes down to this: Do you think it would be FUN for your players to be able to use their extra attack during a readied attack, or do you think it would make the game less FUN.
Remember, they are there to have fun. If you think it's too OP, and so do your players, then NO they can't. If the players think it's not OP, but you do, then it comes down to DM's choice. Personally, I want the players to have FUN. You can always fudge the numbers on your end if the bad guys are losing too quickly.
If you read the rules as Christopher and Thyzer shows it, then the Ready action contradict itself.
It says that you will be using your Reaction, thus it won't be considered your Action.
[...] you can act later in the round using your reaction.
Yet, in the next paragraph, it clearly says that you can take the Action you chose on your turn (or move.)
[...] you chose the action you will take in response to that trigger, or choose to move up to your speed [...]
So, if you chose the Attack action and since there it says you can take your Extra Attack, I think it is allowed. At least, that's how we play it at our tables.
The fact that you can move your full speed on the trigger also very much sounds like you have the equivalent of a full action at the time the trigger happens. A standard Reaction does not give you that opportunity.
What the rules insist on is the fact that you have a single Reaction. In other words, you won't be able to react to someone going past you and take your ready action reaction. You have to choose one or the other (if your ready action is to attack when a creature comes close enough, then you should use the readied action, obviously.)