What To Do When The Person Sitting Next To You Won't Stop Talking

Step 1: Sleep

Pretend to go to sleep. To make sure they definitely get the message, put on your MP3 player and your blow-up pillow and your eye mask and your ear plugs. Block Everything Out.

Step 2: The flight attendant

Use the flight attendant to your advantage. Call her over to your seat and tell her to 'get this freak away from me'.
When returning your food tray, advise her to wash her hands after touching it so she doesn't contract the highly contagious virus you picked up during your recent visit to the Amazon.
While you're at it, ask if you can get an upgrade on account of your health problems and the fact you may only have months to live.

Step 3: Props

Even when stuck in your seat, you have several props at your disposal:
Go very quiet, start rummaging around for the sick bag and make loud retching noises into it.
When you're given your food, make sure you throw it all over the place. Start answering back but only when your mouth is full so it sprays all over them.
Pull out the life jacket from under your seat and explain that you're just getting prepared as every plane you've ever been on has crashed.