ABSTRACT. The stepped wedge cluster randomized trial is an attractive design for evaluating health services delivery or policy interventions. In this design, clusters start in the control condition and gradually cross over to the treatment based on a schedule dictated by random assignment. Outcomes may be assessed on the same individuals over time (i.e., a cohort design) or different individuals (i.e., a cross-sectional design). A key consideration in this design is that sample size calculation and analysis must account for within-period as well as between-period intracluster correlations; cohort designs have additional correlations due to repeated measures on the same individuals. While numerous methods have been developed to account for within- and between-period intracluster correlations with a single level of clustering during each time period, few methods are available to accommodate multiple levels of clustering. Our objectives were to develop computationally-efficient sample size procedures that recognize within-period and between-period intracluster correlations in stepped wedge trials with more than two levels of clustering. Focusing on three levels of clustering and assuming equal cluster-period sizes, we consider three variants, depending on whether each level is treated as a cross-sectional or closed-cohort design. We introduce an extended block exchangeable matrix to characterize the correlation structures both within- and between-clusters in each cluster-period and develop convenient sample size expressions that depend on this correlation structure. With a continuous outcome, we show the sample size expression depends on the correlation structure only through two eigenvalues of the extended block exchangeable matrix. For binary outcomes under a mixed effects framework, we develop a sample size expression based on a first-order Taylor approximation. We conduct simulation studies to examine the finite-sample properties of the proposed sample size algorithms and demonstrate the application of the proposed methods using the Washington State Expedited Partner Therapy trial: a multilevel stepped wedge trial that randomized local health jurisdictions (level 4) consisting of clinics (level 3) and observed patients (level 2) with respect to their Chlamydia infection status (level 1).