Tetrapanax.

The trunks on Tetrapanax look very similar, but the leaves appear less glossy and lobed slightly differently. A few sources mention they produce black drupes, unlike the white ones on the plant. Could it be a different cultivar, or an other similar looking genus?

Well I did find images of Tetrapanax "steroidal giant" at the botanical garden I went to, but it's not the same plant I saw. I'm wondering if the "sensitive" I remember seeing on the name plaque was the name of the cultivar or variety, though I can't seem to find any information on it.

Found it.
It's a Fatsia japonica.
Apparently the Tetrapanax used to be included in the Fatsia genus. Also, F. japonica appears to have black fruits as well, but I must have seen it early enough in the year that they hadn't changed color yet. The leaves on F. japonica are an exact match, though I still can't seem to find what the "sensitive" part could have been.

Like ivy, to which Fatsia is closely related, the flowers are a pale cream to yellow and kind of knobby looking, followed by yellow fruit which can eventually ripen to black. What you were looking at was probably just immature fruit.

The 'sensitive' may have been with respect to cold tolerance. Unless in a very mild microclimate, Fatsia japonica would only be marginally hardy in VA and the foliage easily damaged at temperatures into the low 20's. Relatively root hardy if well mulched and like the English ivy, can be cut back hard to regenerate quickly in spring and summer.

This plant sometimes goes by the common name of paper plant, an indication of its close relationship with rice paper plant, Tetrapanax. Tetrapanax is still sometimes listed as Fatsia papyrifera. Nice drawing, btw :-) Very good for ID purposes!!

Yep, I'm quite sure now the fruit was just immature when I saw it. The plant itself appeared to be doing well and had no frost damage on the foliage even after it snowed. We were about 20 miles from the coast, so the air temperatures weren't too cold this time of year; I don't think it got more than a degree or two below freezing most nights. I really can't remember what the sensitive could have been, as the only things the name plaques showed was the scientific and common name. I believe the plaque may have mentioned something similar to papyrifera, since I kind of remember thinking it was interesting, but it could have been another plant I saw that day too. Or, maybe the plant had the wrong species name. I'd have to go back in the spring and see it again. All I know is that the shrub at my friend's house looked more like Fatsia japonica than Tetrapanax, and the one at the botanical garden, whatever it was listed as, looked like the plant at the house.
Anyway, thank you all for the help (and the compliment!). I really liked this plant and didn't think to post here until just recently, as I have been wondering what it was called for the past month or so. I wish it was hardy enough for my zone, I'd take some cuttings home with me from the plant in Virginia. Would it grow well as a potted plant? I could always take it inside for the winter if I had some growing. My friend's mother was planning on pruning the plant sometime, so I might have a chance to save some pieces from going to waste.

Fatsia will tolerate freezing temperatures to a degree - much into the mid to low 20's for any length of time will typically result in foliar damage or even dieback. But it should resprout. Here, where we generally have mild winter temperatures or if the plant is located in protected area, it can easily reach 10-12' and is often cut back hard simply to regenerate and fill in lower bare stems.

In colder areas, this can be grown very successfully as a houseplant. Tolerates low light conditions very well. Just allow for a sufficiently large container - it is not a very petite plant :-)

Grows in much of the UK too. Not quite as striking but a possible candidate if you want a pot plant is the inter- generic Fatshedera lizei which is a cross between Fatsia and Hedera. I have one which I rescued from an office rubbish bin which has grown out in my garden for over 20 years.

Fatsia makes a great houseplant
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This plant is quite large, and it is half of a small plant I bought last summer. The other half is in my dining room LOL.
I'm going to plant it outside in the shade this comiing summer when it's warm.
Keep it in the house as a houseplant, it's beautiful, just don't overwater it.

Here's a picture of one of mine before it got nipped pretty hard by our cold November spell when my garden got down to 9 F. It will recover.
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I have some Fatshedera someplace in the garden. Being crossed with Ivy, it's a strange plant. It doesn't quite make it as a vine or shrub. The growing tips get about 3 feet high before flopping over. Tying them up may be the way to go, but I don't have the will to go to the trouble.
Mike