It's possible to launch Google Chrome in incognito mode on a PC by adding
-incognito to the end of the desktop shortcut. Is it possible to launch to Chrome on an iPad in incognito mode?
Apple does not allow command line access on iOS. This may be possible with a jailbroken device, but I have never investigated that.
I looked for a setting in Chrome to allow this and I could not find one on the Desktop settings panel, or in the advanced settings. My thought was that the setting would be synced to iOS and the iOS app would honor that request.
The last thing I can think of is to be in an incognito tab when closing the app, that way it will restore that tab when restarting the app. This worked in my testing, although it remembered and reloaded the page it was on.