In article , en? [Howd E.
Doodat] wrote...

I have a tree (or weed or some kind of plant) growing out back of my
house. I've only seen one other like it in my life, and that was about
20 years ago, so they are either very uncommon or they are constantly
cut back by homeowners (although I've NEVER seen one in the woods).

Where do you live? That's often relevant to these questions.

Basically it has a single stem or trunk, although there are a couple
of minor offshoots coming off the base.

In its first few weeks in June, it grew roughly 6" per DAY! In the
last month it has only grown another foot and now stands about 15'.
What is more remarkable here is that its root structure HAS to be
horizontal: it is growing out from under a pile of roofing shingles
and tree branches that I discarded on a concrete slab.

The main stem/trunk is about 2" in diameter, woody at the base but
quickly becoming green about a foot or 2 up from the bottom.

So, it's an actual woody plant, not a giant herbaceous growth like:

http://www.poisoncentre.be/Nl/Heracl...egazzianum.htm
http://www.agf.gov.bc.ca/cropprot/gianthogweed.htm
http://whatcom.wsu.edu/ag/homehort/weed/hogweed.htm

About every foot or so are 3 "branches" radiating from the trunk at
roughly 120 degrees. These "branches" each terminate in a single leaf.

Sounds like whorls of three leaves.

The leaf has a shape something like gum, but the lobes are not as
pronounced.

You mean, slightly palmately lobed, something like Liquidambar [sweet gum]?

http://botany.cs.tamu.edu/FLORA/swts/hama003.jpg

The leaf is fuzzy on the bottom side, and the top tends
toward shininess although it also has a slight fuzz. It's pea green on
top and lighter on the bottom.

Now here is the clue that makes this tree unique: most of its leaves
are in excess of 20" across!

Might well be a seedling/sapling of _Paulownia tomentosa_. They can
grow very fast and do often have very large juvenile leaves. The
leaves are usually just in opposite pairs, but they can be whorled
like yours, especially on vigorous shoots. It eventually becomes a
branched woody tree with smaller leaves and ornamental flowers:

http://www.paulownia.com/
http://www.toadgully.com.au/are_paulownia.php
http://www.nps.gov/plants/alien/fact/pato1.htm
http://www.mediterraneangardensociet...tomentosa.html
http://www.wb2020.qld.gov.au/icm/baf...s/pawlonia.jpg
http://python.ex.ac.uk/ead/be/species/p/paulownia/
http://www.paulownia.it/
http://www.forestry.auburn.edu/samue...press_tree.htm
http://www.emilycompost.com/empress_tree.htm

Superficially similar trees are in the genus Catalpa [not actually
a very close relative of Paulownia], and they do have leaves in whorls
of three. The leaves can also sometimes have a few lobes. I don't recall
seeing any such vigorous, very-large-leaved saplings or suckers as
Paulownia often produces, but I suppose it's possible:

http://project.bio.iastate.edu/trees...pa_leaves.html
http://botany.cs.tamu.edu/FLORA/dcs420/fa06/fa06056.jpg
http://www.ces.ncsu.edu/depts/hort/c...nonioides.html

[snip]

cheers