I don't have that tree, but I think it's aesculus pavia, red horse chestnut. I saw some in bloom yesterday, very pretty. Check out the lower picture in the link, it looks like it gets huge!

Here is a link that might be useful: Red Horse Chestnut

That is a big one. I don't think I've seen one get more than 20 feet around here. When it was installed I was told no more than 30 feet tall and a slow grower.

You see them here and there, but I've never spoken to anyone who actually has one.

Common names and the confusion they sometimes cause...
A.pavia is the Red Buckeye, native to the U.S., and the photos Jeanne linked are pretty representative.
Red Horsechesnut, however is A.X carnea, a hybrid of A.pavia X A.hippocastanum, the European Horsechestnut. These come fairly 'true' from seed, and there are several named selections available in the nursery trade - Briottii, Ft. McNair, O'Neill's Red, etc. Some have red flowers, others pink or pink with a yellow throat. All very pretty in bloom.

Here is a link that might be useful: A.x carnea - Red Horsechestnut

We were admiring a red buckeye here on campus (ohio State) this afternoon. There is a beautiful large white buckeye just outside my building, and a red one just down the street.

Somehow, I never noticed the beautiful blooms on it before this year, even though I have grown up with buckeye trees...

Very nice tree, very pretty stonecrop next to the lamb's ear. :-)

Your clematis looks much like my "Ramona"; it's hard to tell colors from photos but since you described it as purple I'd have to say that's not it, as mine is a tad bluer. I bet the folks over on the clematis forum would ID it in a heartbeat.

Lucky is right, of course. Red Buckeye and Red Horsechestnut are not the same tree. I even saw an online nursery cofuse the two, so you need to be careful in making sure they are identified and labeled correctly, or you may end up with something you didn't intend to plant.

Propagation of Red Buckeye is about as easy as it gets. The nuts are ready for planting immediately after they fall from the tree. If they wait too long, it decreases their chances of germination. Ohio Buckeyes need cold stratified, however.