This is what I have found that works: 1) use the "back arrow' button (<--) at the top left on the Roku remote to turn on the Roku. It will remain on until it detects there is no activity. Or, 2) 'You can turn off your eco mode settings by going to settings and switch off the low power mode." https://homex.com/ask/how-do-you-turn-off-eco-mode-on-roku
Other Roku users in this forum have posted about using method 2.
@milchuck, what model Roku do you own? I was originally posting to a Roku Ultra forum where lots of people were having an automatic shut-off problem. If it is more than a few years older than a 2019 Roku Ultra, it may indeed not have the 'Auto power savings' function. However, I also tried my old Roku Ultra (2011) remote's back arrow button and it turned on my new, ten-day old Roku Ultra. See below for the Power save function. My old Roku does not have an 'Auto power savings' function, but it is irrelevant because I can turn on the new Roku without fail using the back arrow using either the new or the old remote.
maybe this will help. I just bought my Roku Ultra. It's fabulous EXCEPT ... I noticed that after a period of inactivity, or after I shut my TV off for a few hours, when I tried to activate my Roku, it was turned off! I too would up unplugging and plugging the power cord in and having to wait until the device booted up again. I finally checked the settings and discovered a Power Saver function I hadn't been told about when I first set up the Ultra. When enabled [checked], the device will power off automatically after 20 minutes of inactivity. The Ultra has this option alone among all other Roku devices because it has a fan. After experimenting a little, I found I can activate the device by simply pressing the Home button and not having to disconnect and reconnect the power source. The automatic Power off function appears to be the default setting.
It would be nice if Roku had explained this in the materials that came with the Ultra when I bought it from Best Buy.