We construct infinite-dimensional Banach spaces and infinitely generated Banach algebras of functions that, except for 0, satisfy some kind of special or pathological property. Three of these structures are: a Banach algebra of everywhere continuous bounded functions which are not Riemannintegrable ; a Banach space of Lebesgue-integrable functions that are not Riemann-integrable; an algebra of continuous unbounded functions defined on an arbitrary non-compact metric space.