We report on TeV Delta *g-ray observations of the blazar Mrk 421 (redshift of 0.031) with the VERITAS observatory and the Whipple 10 m Cherenkov telescope. The excellent sensitivity of VERITAS allowed us to sample the TeV Delta *g-ray fluxes and energy spectra with unprecedented accuracy where Mrk 421 was detected in each of the pointings. A total of 47.3 hr of VERITAS and 96 hr of Whipple 10 m data were acquired between 2006 January and 2008 June. We present the results of a study of the TeV Delta *g-ray energy spectra as a function of time and for different flux levels. On 2008 May 2 and 3, bright TeV Delta *g-ray flares were detected with fluxes reaching the level of 10 Crab. The TeV Delta *g-ray data were complemented with radio, optical, and X-ray observations, with flux variability found in all bands except for the radio wave band. The combination of the Rossi X-ray Timing Explorer and Swift X-ray data reveal spectral hardening with increasing flux levels, often correlated with an increase of the source activity in TeV Delta *g-rays. Contemporaneous spectral energy distributions were generated for 18 nights, each of which are reasonably described by a one-zone synchrotron self-Compton model.