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calling maximize recursively

Hello all.

I am trying to call an instance of Maximize[] within Maximize[].

Essentially I'm trying to perform a traditional Bellman's equation
optimization, where you maximize value[now] + value[later], and
value[later] is itself another maximization, and so on.

Here is a very simplified example of a recursive Maximize[] call. If
you can figure out a way to get this to work without deleting one of
the two Mazimize[]s then you'll save the day for me! (I am aware this
particular problem could be done far easier, but it's the setup here
that matters, not the particular problem.)

function1[x_] := x - x^2;

function2[bound_] := Maximize[ {function1[x], x<bound} , x ];

Maximize [ function2[bestbound] , bestbound ]

(if you import this in to mathematica it gives an error, because it
tries to pass bestbound as a pure variable to function2, rather than a
candidate value of bestbound to function2)

Please help,

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