Tosfos Shabbos 40b D.H. ושמע מינה כלי שני אינו מבשל points out that a keli sheni will not allow cooking because

אין דופנותיו חמין והולך ומתקרר its walls are not hot and so it will cool off.

Bearing this reason in mind, what will be involved in keeping a keli sheni or keli shelishi warm on top of the Shabbos kettle in order to have a hotter drink on Shabbos.

  • It still is not warm enough to cook on its own. – N.T. Mar 9 at 18:36

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