I've recently stumbled on George Dvorsky's post on io9 (http://io9.com/5945801/8-philosophical-questions-that-well-never-solve). He brings forward 8 such questions:
1. Why is there something rather than nothing?
2. Is our universe real?
3. Do we have free will?
4. Does God exist?
5. Is there life after death?
6. Can you really experience anything objectively?
7. What is the best moral system?
8. What are numbers?
For natural scientists Question 2, 6 and 8 seem to be particularly disturbing if they cannot be answered. In this case, we have to rely on assumptions here.
My query to you is: Are there any other such fundamental questions that we have to leave unanswered?