Less privileged students disproportionately aim low with their university applications and often fail to apply to the institutions that match their academic qualifications. Little attention, however, has been directed toward the converse phenomenon, that of aiming high. These kinds of applications are commonly called 'reaches,' meaning an applicant's credentials fall slightly below the institution's range for the average first-year student. Using nationally representative survey data from the United States, this study examines whether social background predicts the likelihood of applying to reach institutions and assesses the consequences for disparities in enrollments. We find a strong relationship between students' socioeconomic background and their likelihood of applying to reach institutions, even after controlling for a range of academic and non-academic characteristics. Further, there is a substantial pay-off to applying to reach institutions, revealing the degree to which these class-based choices at the application stage contribute to the social stratification of higher education.