Student or Learner
"He changed so much that you may..... not recognize him"
I put "as well" in but the key state it's supposed to be just "well" (it was a multiple choice question so there can be only one answer).
So what is the difference between may as well and may well?
"may well" is simply an intensification of 'may'.
The meaning of 'may' here lies on the spectrum of possibility, rather than permission.
"You may well not recognise him" means almost the same as "You may not recognise him". But since he has changed so much, the speaker has intensified the possibility that you won't recognise him by inserting 'well'. This is used almost with the same meaning as 'very'. In fact, the speaker could exaggerate further and say, "You may very well not recognise him".
"may as well" means "It would be just as well if ...; You might just as usefully ..."
Now, you can see that this doesn't fit the meaning of your sentence.
The meaning of 'may' here relates to permission or advisability of doing something, not with possibility.
Some examples are:
"You've been studying for 8 hours and you're worn out. You're too tired to learn anything. You may as well stop". That is, it would be just as sensible for you to stop as to keep studying.
At a cricket match: "It's raining hard. We may as well go home".
So, now taking an example where 'may' is ambiguous:
"The employees may stop working".
This could mean either that i) it's possibile that they'll stop working; they might go on strike, or ii) They have permission to stop working.
Inserting either of the above two phrases disambiguates that sentence.
"The employees may well stop working" means i.
"The employees may as well stop working" means ii.
Since your original sentence deals with possibility, you'd need to choose "may well".