The TeV binary system LS I +61 degrees 303 is known for its regular, non-thermal emission pattern that traces the orbital period of the compact object in its 26.5 day orbit around its B0 Ve star companion. The system typically presents elevated TeV emission around apastron passage with flux levels between 5% and 15% of the steady flux from the Crab Nebula (> 300 GeV). In this article, VERITAS observations of LS I + 61 degrees. 303 taken in late 2014 are presented, during which bright TeV flares around apastron at flux levels peaking above 30% of the Crab Nebula flux were detected. This is the brightest such activity from this source ever seen in the TeV regime. The strong outbursts have rise and fall times of less than a day. The short timescale of the flares, in conjunction with the observation of 10 TeV photons from LS I + 61 degrees 303 during the flares, provides constraints on the properties of the accelerator in the source.