The Maximum: Would be infinite, no? Say you did each peak by walking from your home. Or from China? You decide.
Or: start anywhere. Walk half distance to peak. Take gratuitous detour, get lost, wind up back at start. Walk 3/4 of distance to peak. Detour back to trailhead. Walk 4/5 of distance to peak. Detour back to trailhead. Continue in this pattern...
I think the question assumed a finite set of acceptable trails (including the Owl's Head slide and maybe a few of the "standard bushwhacks"), which implies a corresponding finite set of acceptable trailheads. If you count distance by redlining, the maximum distance is the combined length of all the trails. (You'd have to plan your itinerary carefully if you're obeying the AMC rules, or you might accidentally end by reaching a trailhead after your last summit, without having redlined some other trail(s) to the same mountain from a different trailhead.)
You could constrain the maximum further with additional rules, such as:
- visit each peak exactly once
- no trail section may be traveled more than once (with the exception of returns along summit spurs). Thus, no loops and fewer silly branches.
Note either of these rules would make it impossible to "do the peaks one at a time" using maintained trails. (West Bond, for example...)
What you really want is some way of saying "no crazy side trips" - not sure how to say that while still permitting not-quite-crazy ideas like doing Carrigain "the back way" (via Desolation Tr). I guess you could specify acceptable trails *for each peak*.