OK, so Shakespeare and the Bible are in automatically. Now you have 3 more that you can take, and "How to Build a Raft and Get Off a Deserted Island" isn't one of them. And, no, you don't have a piano, so "The World's Biggest Fake Book" is kind of a waste.
Here are my three:
"Lord of the Rings" for sheer enjoyment, fantasy, and escapism. As soon as you finish it, you can start again, and the adventures and language never pale.
Dante's "Divine Comedy." T.S. Eliot said that Shakespeare and Dante divided the world between them--there was no third. The more I read it, the greater the depth, symbolism, ethical thought, etc. This is truly a book for the ages; it would never pale.
"Ulysses" by James Joyce. It was a tough call to pick this over "Remembrance of Things Past" (another daunting read), but Joyce is in English, not translation, and his sheer abundant inventiveness, word play, character study, and complexity could keep you going for years.
So that's it for me: 5 books for the LOOOONNNNGGG stay on the island.
Oh, and maybe a "Field Guide to Edible Plants on Deserted Islands" might come in handy.

I'd take "Ulysses" and "A La Recherche du Temps Perdu" ("Remembrance of Things Past" in the original, since I'd have all the time in the world to figure out the French without a dictionary). Also, I'd take "Laszlo Polgar's Chess: 5334 Problems, Combinations, and Games" because the darn thing is big enough to use as a raft.

A book by George Orwell, a book by Franz Kafka, and maybe something by Proust (never read him but have heard fantastic things about his work). Maybe War and Peace, but I fear that even with a lifetime of doing nothing I still might not finish it!

Lol, the Polgar chess book. I used to own a copy myself, but [remarkably] seem to have lost it!!!

War and Peace is actually a faster read than you'd think.

Take 'Brave new world' of Aldous Huxley with you and feel if you are in the reservate.

Hey, if you're on an island, school's out, right? I'd take good reads instead of daunting homework assignments--there are other things to do on islands besides have a nervous breakdown. A Christmas Carol, To Kill a Mockingbird, Good-bye Mr. Chips...or Different Seasons, The Old Man and the Sea, and Boswell's Life of Johnson. Things you can actually get lost in and enjoy, without getting stuck on one paragraph and forcing yourself to plow through it over and over without really absorbing it. I mean, does anybody actually CURL UP with Ulysses?

Actually, I do curl up with Ulysses. But I did my homework first -- wrote my master's thesis on it 30 years ago. I'm currently listening to the audio book and following along with the paper edition.

I stand corrected. Here's at least one Ulysses curler-upper for the record, if we don't count Joseph Campbell, who's dead. Rock on, Bach.