To solve the problem of finding the maximum value of ((1+a)(1+b)(1+c)) given (a, b, c > 0) and (a+b+c=1), follow these steps:

Step 1: Symmetry Insight

For symmetric expressions under the constraint (a+b+c=1), the maximum often occurs when (a=b=c). This is a key observation from AM-GM inequality and symmetric function properties.

Step 2: Equal Variables Case

If (a=b=c), then (a=b=c=\frac{1}{3}) (since (a+b+c=1)).

Step 3: Calculate the Expression

Substitute (a=b=c=\frac{1}{3}) into ((1+a)(1+b)(1+c)):
[ \left(1+\frac{1}{3}\right)^3 = \left(\frac{4}{3}\right)^3 = \frac{64}{27} ]

Verification

Using Lagrange multipliers or derivative analysis confirms that this is indeed the maximum (as shown in the thought process).

Answer: (\boxed{\dfrac{64}{27}})

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