Also, if it is a weed, I do realize how comical it must look to experienced gardeners that I've been "babying" this thing for so long when I could have planted some nice annuals there! :P

I'm pretty sure those are not strawberries. But then again, the flowering bit does look like a new strawberry.

Those are definitely not strawberries but may be a flower of some kind because I see a flower bud. Strawberries do not grow tall like that and that's sure not a strawberry leaf. What flowers did you have in that planter or near by. A stray seed. The leaf looks much like st. john's wort

Maybe Duchesnea.

OK! I guess as long as it's not the strawberries I'll go ahead and yank it out...whatever it is I don't want it there...

Definitely Rosaceae. Neither Strawberry nor Duchesnia with those long flowering stems. Looks a Potentilla species. An open flower picture would help.

I was thinking Potentilla too.