Can a negative pH exist?
Yes, pH can be negative.

The official definition of pH is: negative log of H+ activity, not negative log of H+ concentration, however.  See pH Paradoxes: Demonstrating That It Is Not True That pH ≡ -log[H+] J. Chem. Educ., 2006, 83 (5), p 752.

By either definition pH can be negative in concentrated solutions of strong acids.  For example 5M HCl would have a negative pH.