Pulse compression in a differentially pumped neon-filled hollow fiber was used to generate high-power few-cycle laser pulses. The pulse compression process was optimized by adjusting gas pressure and laser chirp to produce the shortest laser pulses. Precise dispersion control enabled the generation of laser pulses with duration of $3.7\mathrm{fs}$ and energy of $1.2\mathrm{mJ}$ . This corresponds to an output of $1.5\text{cycle}$ , $0.3\mathrm{TW}$ pulses at a $1\mathrm{kHz}$ repetition rate using positively chirped $33\mathrm{fs}$ laser pulses.