You should be able to create a contact "me" which is then searched as you type "me" in the To: field when composing an email. You will then need to pick the appropriate "me" contact from the drop-down list of contacts that contain the characters "me". You can't simply type "me" in the To: field and expect it to be routed to you.
I have an "@ME" contact for the same reason. I prefix it with "@" so that it is at the top of my contacts list and no other real contacts contain the "@" character in their name - so the drop-down list is very short.
I have read that Gmail treats email that you send to yourself as duplicate(?) and consequently it does not appear in your Inbox - only in your Sent folder. Although this does not happen to me - it appears in my Inbox like any normal email.
EDIT: I have found that after closing and opening Gmail it will no longer prompt with the "me", "myself" or "@ME" contacts in the autocomplete dropdown when filling in the "To" field! This is bizarre. Changing the contact name in some way - such as adding a real name eg "@ME (Joe Bloggs)" resolves the issue temporarily, but it disappears again when returning to Gmail later. Contacts where I have used my full name (without any funny prefix) remain in the autocomplete dropdown.