Is there a way to do this?
The long answer is no.
The short answer, which I prefer, is nope!
Anchor points in Illustrator have only one in and one out. What you want to do would require an anchor point to have two ins and one out or two outs and one in. The math would be too confusing. To stop AI from deleting the added anchor, check the Disable Auto Add/Delete in your preferences.
Thanks for all the quick answers. At least I know I'm not crazy for not finding it then. Thanks for the help!
The math would be too confusing.
I think it's more a matter of traditional convention in drawing programs than anything else. Consider:
Flash does indeed allow you to have a single path with more than two segments connected to a single anchorPoint. For example, in Flash, use the Line tool to draw a tic-tac-toe grid. Get the white pointer and select the path. You'll see that it's treated as "one path." Select one of the intersection points and move it. All four of its associated segments are affected.
Illustrator, Draw, FreeHand, Canvas, et all, draw cubic Bezier curves, which are defined by four coordinate pairs each (i.e.; a single segment). It's just the interface of the program that strings them together by having the segments "share" one of the coordinate pairs.
I'm not saying I'd necessarily be in favor of it; that would depend upon how well the interface was pulled off. But I don't think there's any mathematical reason at all why cubic Beziers couldn't be treated the same way. They just aren't in the traditional Bezier program interface.