It depends. Some services will automatically start the next episode after the current one ends. Generally, if you're watching a movie, once it ends nothing will automatically start. If you're watching a "live" channel, it will stream indefinitely. Roku devices (other than TVs) don't turn off. Any power button on the Roku remote is strictly for the TV. There's a "Bandwidth saver" option in the Network settings that will stop streaming after four hours if no button on the remote is pressed.
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