We present calculations with subcycle precision of laser-driven filamentation in argon which show that the laser peak intensity can exceed the clamping intensity by a factor of 3 during recurring spikes which can last over several centimeters of propagation. The high intensity occurs during a few-femtosecond subpulse in the trailing edge of the main pulse and gives rise to isolated 500 attosecond, 2 pJ pulses which can be extracted from the filament. We also show that the high harmonic radiation emerging from the filament is an excellent diagnostic of the intensity spikes.