Countdown number game: Number of targets

The number game, as seen in the UK tv program countdown, involves a
set of 6 natural numbers, and a 'target' number (also a positive
integer). The goal is to form the target number using the 6 numbers
(each being used once only in the final expression), and the
arithmetic operations addition, subtraction, multiplication and
division. (As well as parentheses and so on.)

e.g. (2+3)/5 + 9*25 - 1 is a possible solution for the target number
of 225.

The question is: What is the maximum number of distinct possible
targets that can be formed by any given set of 6 integers? (We should
ignore the gameshow bounds of 1-999 on the target number, I think,
just to make the question interesting? Possibly for convenience, we
should allow non-integer targets?) What sort of set of integers can
attain this upper bound?

I mean, as a gross bound, one can consider the process of taking a
pair from the 6 integers, applying an operator to them, then taking a
pair from the remaining numbers, and apply the same process on the new
set of 5 numbers etc, ending up with 4^5*prod_{i=2}^{6}(i c 2)
possibilities. But clearly lots of these will be the same, due to
commutativity of operations.

Does anyone have any ideas?

.

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