There has been much discussion as to what little Prince George's last name is or will be. Some say "Cambridge." Some say "Wales." Some say "Windsor." Some say "Mountbatten-Windsor."
It's this last one -- Mountbatten-Windsor -- that most needs to be addressed.
There are only two people alive today who can be called Mountbatten-Windsors in any statutory capacity -- Prince Edward's children (the Queen's grandchildren), Louise and James.
Yes, various other royals have been known to use "Mountbatten-Windsor" as their surname. These same royals could also affix to themselves the surname "Spaghetti," if they so desired, but there's nothing statutory about either.
On February 8th, 1960, Queen Elizabeth II declared the following in an Order in Council:
"I declare My Will and Pleasure that, while I and My Children shall continue to be styled and known as the House and Family of Windsor, My descendants other than descendants enjoying the style, title or attribute of Royal Highness and the titular dignity of Prince or Princess and female descendants who marry and their descendants shall bear the name of Mountbatten-Windsor." (from The London Gazette 9th February 1960, via Heraldica.org)
Translated, this means that only the Queen's non-royal male-line descendants (and their unmarried daughters) are to be recognized as bearing the surname "Mountbatten-Windsor."
Prince Edward, however, declined to have the "titular dignity of Prince or Princess" bestowed upon either of his children. By doing so, he's made it so that his daughter Louise and his son James can be considered the first statutory Mountbatten-Windsors.
All of the Queen's other male-line descendants (and their unmarried daughters) are princes and princesses, thus making each one a "Windsor" -- including, Princes Charles, William, Henry (Harry), and now little George.
All this could change, of course, once a new monarch is on the throne. Either Charles, William, or George could decide to officially proclaim the royal house of the United Kingdom to be "the House of Mountbatten-Windsor" -- and that will be that. Much less likely, any one of them could even choose to declare the royal house "the House of Oldenburg," as Charles, William, and George are all patrilineally descended (via Prince Philip) from the Oldenburg kings of Denmark (and then Greece).
But until that day, no matter whether you hear him called -- "George Cambridge," "George Wales" or "George Mountbatten-Windsor" -- the closest thing to a statutory surname that our new little prince will have will be "Windsor."