I agree, Smilax or something similar. Those fruits look too juicy and large. Here is a link that might be useful: Smilax fruit

Here are more photos, sorry, I know that they are large but they might make it easier to identify if anyone might know what it is. Here is a link that might be useful: More Mystery Tree Photos

I may be way off but I think it may be a Redbay tree. Rub the leaves and see if you get the smell of bay. It is a member of the Laurel family and is being threatened in Ga, North Florida and the Carolinas by the Ambrosia Beetle. You may notice some leaves that have a funny curl part to them. If so this is a Gall of somesort but it won't hurt the tree. If it isn't Redbay then it still may be a part of the Bay family. We planted a Redbay as a host to Spicebush/Palamdes Butterfly. But afterwards discovered that they prefer the Camphor trees that come up as volunteers in our yard. (I know they are invasive but we keep one or two trimmed back so that they have new growth to lay eggs on.) If you do have a Redbay or other member of that family, there is a group in Ga that collect the seeds in the fall to plant to get seedlings to replace the ones destroyed by the beetle. I ended up sending some seeds to a guy in Houston and he grew some to plant in the last native plant park inside the loop around Houston. They are a native plant there but they are very hard to find as seedlings. Hope this helps. Let me know if I am right.

It's called Grecian Windflower, or just Windflower. Were there any other colors besides the blue ones? I've seen them bloom in pink, white, yellow and lavender too. They grow from a small bulb, similar in size to grape hyacinth.