Got it--thanks to Chris Fox at Esri. Use the Spatial Join geoprocessing tool, make the smaller polygons the Join Features, the encompassing polygons the target, choose One-to-One (so you only have one row per target feature). For Match Option, choose CONTAINS. You will get a count field in your output feature class. And if you want statistics on any join attributes, for instance if you want to see a sum of an attribute for all join polygons contained by a target polygon, in the Field Map right-click the attribute and change the merge rule. Voila!