Yeast telomeric DNA is assembled into a nonnucleosomal chromatin structure known as the telosome, which is thought to influence the transcriptional repression of genes placed in its vicinity, a phenomenon called telomere position effect (TPE). The product of the RAP1 gene, Rap1p, is a component of the telosome. We show that the fraction of cells exhibiting TPE can be substantially reduced by expressing large amounts of a deletion derivative of Rap1p that is unable to bind DNA, called Rap1 delta BBp, or by introducing extra telomeres on a linear plasmid, presumably because both compete in trans with telomeric chromatin for factor(s) important for TPE. This reduction in TPE, observed in three different strains, was demonstrated for two different genes, each assayed at a different telomere. In contrast, the addition of internal tracts of telomeric DNA on a circular plasmid had very little effect on TPE. The product of the SIR3 gene, Sir3p, appears to be limiting for TPE. Overexpression of Sir3p completely suppressed the reduction in TPE observed with expression of Rap1 delta BBp, but did not restore high levels of TPE to cells with extra telomeres. These results suggest that extra telomeres must titrate a factor other than Sir3p that is important for TPE. These results also provide evidence for a terminus-specific binding factor that is a factor with a higher affinity for DNA termini than for nonterminal tracts of telomeric DNA and indicate that this factor is important for TPE.