Silly Physics question...

Multiple times I've seen the claim that the function of the Higgs boson is to give particles their mass. However, virtually all particles I know of are described as having mass (offhand I can think of only two I have heard described as massless: photons and neutrinos, and the latter I've heard may not be entirely massless).

So does that mean that every particle out there (minus those few) is actually a particle group -- the named particle plus one or more Higgs bosons?