One application (which I used to work on) is generating electric field strengths beyond the 'Schwinger limit'. This is a field strength below which various QED processes are exponentially damped, such as electron-positron pair production. Generating such field strengths in the laboratory (as opposed to, say, the surface of pulsars) will allow studies of strong-field QED.

Actually no, the "photonuclear" region is actually already achieveable, the issues in that field are more mundane/engineering related, often how to increase efficiency, how to make things repeatable instead of 5 shots per week, etc. 1e25 W/cm^2 is approaching the next frontier, the strong field quantum regime.