Self-compression and spectral supercontinuum (SC) generated by filamentation of femtosecond laser pulses with duration from 45 fs down to 6 fs in argon gas have been numerically investigated. A 45-fs pulse can be self-compressed into a few-cycle pulse with duration of 12 fs at the post-filamentation region. By properly employing a high-pass filter to select the broadening high-frequency spectra which are almost in phase, the pulse can be further shortened to about 7 fs. By contrast, a 6-fs pulse cannot be further self-compressed into a shorter pulse by filamentation although it can generate much broader SC extending from 200 nm to 1300 nm. It is also found that a separate and strong SC in the ultraviolet (UV) region extending from 220 nm to 300 nm and peaked at about 255 nm can be generated at proper propagation distances, which corresponds to a pulse with duration of about 5 fs.