Is there a way to extend the time you can talk to Siri for?

You can prevent Siri from cutting you off in the command mode by holding down the Home button until you're done speaking.

Normally you activate Siri in the command mode by holding the Home button until Siri responds, then releasing the button and talking. In that mode she'll cut you off pretty quickly. Continuing to hold the Home button bypasses that limitation.

If you say, “Hey Siri, open Notes,” and have a hand free to tap the Compose icon, then tap the Mic button beside the space bar in the Compose keyboard. Immediately Siri will transcribe indefinitely, until you reach over and tap Done. Unless the screen were to shut off, this is a 3-touch solution.

By means of the Shortcuts app that Apple has released more recently than the other answers on this question, you can make a simple chain from “Dictate Text” into “Create Note”, but you still only have the options to stop it with a pause, a short pause, or a tap, not with a codeword, so for your purposes it’s still a 1-touch solution, not safely handsfree.

Edit: iOS 13 broke this so badly it often stops recording after less than a second's pause between words. However a new answer is possible through the new Voice Control utility iOS 13 has introduced under the Settings > Accessibility menu (scroll down into Physical And Motor section). After downloading and activating it, you can now speak to the Home screen, "Open Notes," then, "Click New Note," and now there are a number of commands it can recognize as ending the continuous dictation by doing things with the text. Details at link:

https://support.apple.com/en-us/HT210539

"When the cursor is in a document, email message, text message, or other text field, you can dictate continuously."

I came across this question while trying to setup an iOS Shortcut to dictate a note. In that context, the short pause in Siri is not helpful.

Shortcuts does have a (long winded) work around for this involving repetition and if statements.

This checks if the Dictated Text ends with a '.', and if not, checks for a new Dictated Text and appends the new text, each time looking for an ending '.'.

This doesn't actually extend the wait time, but does allow you to keep speaking, appending your text each time, until you choose to end the note with a '.'. It repeats 3 times, but you could extend this to 5 or more if you wanted.

I've attached the full shortcut, if it helps anyone

There are 2 ways for Siri to tell when you are done dictating an item, a standardized pause, or a tap on a button. If Siri is creating the message, there is no button, so no other way to stop, other than listening for a pause.

After creating a message, you can try appending more to it by saying "Add ...".