Erich Friedman: this is about as long as i could get with a reasonable ending:
six, one, eight, five, two, seven, six, one, fourteen, two, four, four, seven, one, nine, two, three, two, four, seven, seven, one, four, one, two, three, three, sequence.
Eric A.: draw a thin vertical stroke after each letter "e". Count the letters between the strokes. Enjoy!
... Here we go to infinity thanks to the morphism: FOUR --> NINE NINE --> FORTY-FIVE FORTY-FIVE --> FOUR FOUR FOUR FOUR FOUR FOUR FOUR FOUR FOUR FOUR FORTY-FIVE E = SIX TWENTY FOUR FOUR FORTY-FIVE NINE NINE FOUR FOUR FOUR FOUR FOUR FOUR FOUR FOUR FOUR FOUR FORTY-FIVE FORTY-FIVE FORTY-FIVE NINE NINE NINE NINE NINE NINE NINE NINE NINE FOUR FOUR FOUR FOUR FOUR FOUR FOUR FOUR FOUR FORTY-FIVE ... Best, É.