We consider the existence of countably many positive solutions for nonlinear n th-order three-point boundary value problem u ( n ) ( t ) + a ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = α u ( η ) , u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = β u ( η ) , where n ≥ 2 , α ≥ 0 , β ≥ 0 , 0 < η < 1 , α + ( β − α ) η n − 1 < 1 , a ( t ) ∈ L p [ 0 , 1 ] for some p ≥ 1 and has countably many singularities in [ 0 , 1 / 2 ) . The associated Green's function for the n th-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearity f which yield the existence of countably many positive solutions by using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone.