Yes, absolutely. There are at least two categories that are present in most if not all metro areas:
1. Direct suburbs of satellite cities.
This is pretty self explanatory. Most large US cities have other smaller cities on their fringe, which are gradually becoming suburbs of the main city, but which have suburbs of their own. For example,
Frederick, MD is a satellite city of Washington, DC. It was once independent, but is now effectively a suburb. Frederick has its own suburbs, such as
Walkersville, MD. With mayyybe one or two exceptions, you can find places like this surrounding all large US cities. Denver > Boulder > Gunbarrel. Dallas > Denton > Sandy Shores. Atlanta > Marietta > West Hampton.
2. Residential suburbs of mixed-use activity centers.
All metro areas have things called "activity centers." You might also call them "edge cities" or "suburban downtowns." These are suburban places that are clearly within the main metro area, but which have a lot of jobs, with office towers, shopping, etc. Picture a shopping mall with a cluster of high rises in the same area. These places often have residential areas nearby that are clearly all part of the same overall metropolis, but which rely on the more local activity center/edge city for most of their services, and could be called suburbs of the activity center. Think DC > Tysons Corner > Great Falls, or Denver > DTC > Greenwood Village, or Dallas > Las Colinas > Valley Ranch, or Atlanta > Perimeter Center > Dunwoody.