In　the　present　paper　the　author　investigates　the　generalized　nonlinear　initial-boundary　Riemann　problem　with　small　BV　data　for　general　n　x　n　quasilinear　hyperbolic　systems　of　conservation　laws　with　nonlinear　boundary　conditions　in　a　half　space　{(t,　x)vertical　bar　t　＞=　0,　x　＞=　0},　where　the　Riemann　solution　contains　rarefaction　waves.　Combining　the　techniques　proposed　by　Li　and　Kong　with　the　modified　Glimm＇s　functional,　the　author　obtains　the　almost　global　existence　and　lifespan　of　classical　discontinuous　solutions　to　a　class　of　generalized　nonlinear　initial-boundary　Riemann　problem,　which　can　be　regarded　as　a　small　BV　perturbation　of　the　corresponding　nonlinear　initial-boundary　Riemann　problem.　This　result　is　also　applied　to　the　problem　of　planar　steady　supersonic　Euler　flow　past　a　wedge.　(C)　2018　Elsevier　Ltd.　All　rights　reserved.