Your plant appears to be a kalanchoe.  It's a great easy-to-care-for succulent plant often used as a  houseplant.  It has lovely flowers.  Here is some information on caring for it so you can encourage yours to bloom: http://www.clemson.edu/extension/hgic/plants/indoor/flowering/hgic1563.html

Thanks so much for the information and for taking the time to respond. I'll go look into the kalanchoe. I'm very pleased with the quick, and complete, response! What a great service this is!

If I might impose one more time, I've looked at dozens, even hundreds, of pictures of Kalanchoe, but none seem to really be what I have. Mine is very "viney" with long, thin, stiff vines. The leaves are very similar to Kalanchoe, but I don't see any Kalanchoe that have long vines. Most seem upright or bushy.

The plant I have has vines that are probably 1-1/2 to 2 feet long, and are VERY heliocentric. Every single vine twists and turns around to point directly at the window and will even grow "up" for an inch or two before the weight brings it back onto the shelf.

I have looked at a variety of sources, and see there are almost 60 different varieties of Kalanchoe, so I'm hoping my additional description might help narrow down which one I have. Ideally I'd like to snip the one I have as it's a bit leggy and propagate to better pots and soil, but I want to make sure I'm reading the information on the right variety of Kalanchoe.

Thank you ahead of time for any additional help you are able to provide!

And in case it helps, here's the third picture I submitted rotated properly.

Let me ask you this--are the leaves thick and fleshy? Please let me know.

The leaves look very succulent in the photos.  I'm still leaning toward Kalanchoe, because I've seen them get long and leggy like this if they've been allowed to grow and not been cut back when starting to get leggy.  I think your plant has been allowed to grow so long that it cannot hold itself upright.  I think the fact that the ends of the branches try to lift themselves upright in the sun may confirm this.

Please get back to me.  If the leaves are NOT thick and fleshy, let me know.

 If they are, try the propagation techniques--Cut off the last 2-3 inches off a few of the branches, leaving the terminal two leaves.  Allow the cut ends to callous--dry out--for a few days. Insert in moist planting medium, place the pot in a large plastic bag to maintain humidity, and place in a window with indirect sunlight.  This is best done in spring or early summer, but I've done it any time of the year.

Yes, the leaves are pretty thick. I'm not exactly sure what "fleshy" is, but they certainly do seem like a succulent leaf. When a leaf breaks it is pretty full of water. The leaves seem to have many of the succulent characteristics.

The branches/vines have never been able to hold themselves up as long as I've had the plant. It started from a pretty small plant that was a gift, but it's been pretty happy up until the last couple of months when the leaves start turning yellow and getting soft. Eventually the leaves on the older branches turn yellow and soft, then dry up and fall off, causing the legginess. Despite this, the main plant is putting off new shoots, with several more just coming out, so I still think the main plant is relatively happy, but I'm thinking perhaps the pot is too small for it now.

Even the brand new shoots coming from the main stem only grow an inch or two and then they start getting too heavy to stand upright. That's why I say it is "like a vine" (even if it's not a vine plant). Even the new shoots very quickly bend downward toward the shelf. The new vines seem especially happy as they are putting off lots of new leaves and are growing very quickly with no leaves turning yellow or falling off. The leggy vines are much older (i.e., a few years).

I will definitely try out the propagation technique you've described. The very ends of both the older and newer "vines" seem quite happy, and would likely propagate well. Thank you again for all of your time. I definitely appreciate it!

I should probably also note that this plant has never flowered. Perhaps it's not as happy as I think, but it's growing like crazy, but has never flowered, nor do I see anything that looks like it might be buds.

Hmmmm--it's a succulent then...and as far as I know, there aren't any other succulents with leaves like that...I'll keep looking.  If you like, I can put this back in the pool and someone else can take a look at it and perhaps solve your mystery.  Let me know. :-)

Let me ask you this--are you fertilizing it? If you are, it may be too much--this can cause your succulent to get very leggy and weak-stemmed, and may prevent the plant from producing flowers. .  Use a low nitrogen fertilizer at about ½ to ¼ the recommended rate. You may also be able to find fertilizer made for succulents. This component is not completely necessary as too much fertilizer can kill your plant.  

BTW--here's a great article with illustrations for propagating your plant: http://www.wikihow.com/Propagate-Succulent-Plant-Cuttings

No need to put it back in the pool. Your information seems to align well with the plant. I did read an article that says kalanchoe can get leggy and essentially supports your write up about propagation. It even shows cuttings in soil and they look just like the ends of my plant. The site is here:  http://dottypants.blogspot.ca/2015/01/foliage-follow-up-strobilanthes.html

I'm not using any fertilizer on any of my plants. I was thinking of doing it once in a great while, but haven't done it yet.

Thanks again!