What is the best compact survival kit?

Honestly the best thing to do is build your own according to what type of environment you're going to be in. That being said there are a few staples that will work well in any sort of location:

1. Knife and/or multitool. I recommend a stout fixed blade survival knife AND a multitool but if you're on a budget I'd go with a good multitool.

2. Fire-starters; matches, lighters and flint strikers will all work but I would have all of them and keep them in separate pockets just in case. You should also bring some cotton balls and hand sanitizer as both hold a flame well.

3. Signal mirror and emergency blankets. Both can be used as a signal and the blanket obviously as shelter.

4. Paracord. As the name suggests this is rope used in parachuting and is lightweight and strong. It's also cheap and you can find it at most any Army Surplus website.

5. Water purification tablets. You can boil water you happen across but these things will help to kill any bacteria in the water.

These are your basic supplies but I would recommend reading up on the area into which you are venturing and I would also pick up some sort of survival field guide so you know how to properly use the tools you brought with you. And remember to keep your head on straight if you find yourself in a survival situation, panic and despair will kill you quicker than the elements.