Can a 60 Amp Subpanel Feed Another Subpanel?

Yes. The NEC places no limit on how many panelboards can be cascaded off one another. What it does regulate is that each panelboard is protected at or below its rating, each set of feeder conductors is protected at its ampacity, and grounds and neutrals stay separated in every panel downstream of the service.

Panelboard rating: 408.36

Section 408.36 requires each panelboard to be protected on the supply side by an overcurrent device having a rating not greater than that of the panelboard. A panel with a 100 ampere busbar rating fed by a 60 ampere breaker is fine, because 60 is not greater than 100. The reverse is the violation. A 100 ampere breaker feeding a panel rated 60 amperes is not permitted.

So a 100 ampere rated subpanel fed from a 60 ampere breaker in the main is compliant. Installing a 100 ampere breaker inside that subpanel to feed a third panel is also compliant, as long as the conductors leaving that 100 ampere breaker have an ampacity of at least 100 amperes per 240.4, and the third panel is rated 100 amperes or more.

Where the plan usually falls apart

The problem in the question is not legality, it is capacity. Everything downstream of that 60 ampere breaker shares 60 amperes. Putting a 100 ampere breaker two panels down does not create current that the upstream device will pass. The 60 ampere breaker will trip first every time, and it will trip on the whole load rather than on the faulted circuit, which means one overloaded appliance takes out the entire branch of the system.

Selective coordination is not required in dwellings, so this is not a code violation on its own. It is a design problem. If the third panel needs real capacity, the feeder should originate at the main, not two panels deep.

The rules that actually get flagged

Section 408.40 and 250.24(A)(5) are the ones inspectors catch. In every panel downstream of the service disconnect, the grounded conductor and the equipment grounding conductors must be kept separate. That means removing or omitting the bonding screw or strap, landing neutrals on an isolated neutral bar, and landing grounds on a bonded ground bar. Two panels deep, three panels deep, it does not change.

Section 225.30 and 250.32 apply if any of these panels are in a separate structure. Each building gets one feeder unless a permitted exception applies, a disconnecting means is required at the structure per 225.31, and a grounding electrode system is required at that structure per 250.32(A).

Section 110.26 applies to every panel in the chain. Thirty inches of width, thirty six inches of depth, and six feet six inches of headroom in front of each one, with the working space clear and the door able to open at least 90 degrees.

Finally, every panel needs a legible directory per 408.4(A), and where the panel is not in sight of its supply, marking the source and location on the directory saves the next electrician a long afternoon.