Now with some additional rule clarifications from Karthik about using dropped enemy skyrise sections, some interesting heights have appeared.

The normal height for skyrises is up to 60"(7 sections+base) but if the enemy team drops all 7 of their seven sections and you pick them all up and somehow stack them all it adds up to 116" or 9’8"; including the highest score cube this is about 10 feet (basketball hoop height). This is the highest possible valid scoring height.

If we discard scoring, it is theoretically possible to build a tower legally(yes, legally) during a VEX match reaching 770", assuming it somehow does not tip. This involves the following:

Stacking 44 cubes vertex to vertex.
An 8" cube has a diagonal length of 8sqrt(3)= 13.9", but since it has a border radius of 0.875"(1.75"/2), each skyrise cube only has an effective diagonal of
8sqrt(3) - (1.75sqrt(3) - 1.75) = 12.6"
diagonal of cube 8"- (diagonal of cube 1.75" - actually occupied length of 1.75")
Multipled by 44 for a total of 553"

Stacking skyrise sections tip to tip and base to base slightly slanted.
With a bit of slanting the maximum height for one skyrise section is a tiny bit less than sqrt(148), which does not account for the small border radius of the skyrise section rim.
With 14 sections, there is a total height of about 170".

Balancing everything on top of the 47" goal

So if somehow someone finds some way to balance everything on top of each other within the short 2" time frame, the highest possible height that can legally be obtained during a match with scoring objects and the field is about 553+170+47=770" or 64’2".