I remember the episode where Nanny is learning to drive, and she ends up driving an expectant couple to the hospital. There is some confusion, and someone thinks that Nanny was the one having a baby. I remember Butch saying excitedly, "Boy, this is getting neat!"

I recall that this show was part of my Friday "must see TV" when I was a little boy - there was Brady Bunch, Nanny, Partridge Family all in a row!
And in the 69/70 season - The Flying Nun at7:30 (EST) right before the Brady Bunch!

I remember that episode, too, though it's been years since I've seen it
I remember that "Nanny" went upstairs to try on her great-grandmother's(?) wedding gown.
I think Prudence or Butch made a comment about how pretty Nanny would look in the gown and, a moment later, Hal, looking up the stairs said, "Boy, I'll say!" They then showed Nanny in the gown, and she did, indeed, look lovely. The Professor also stared at her and sort of gulped and said quietly, "Your great-grandmother's dress looks just beautiful on youMiss Figalilly." And Nanny said, "Thank you, Professor."
I think the kids were somewhat depressed at the thought of her leaving, and the Professor told them that if they love Nanny they should try to show her how self-sufficient they could be, so she wouldn't feel bad about leaving. This was why Prudence told Nanny "her story" rather than Nanny telling one to her.
Later, Nanny came across a letter her mother(?) had written to her some years ago. It had some advice in it about everything happening at the right time and right place, and following your heart. After reading it, Nanny realized the Everett family wasn't ready to do without her yet (or she without them), so she broke off her engagement to her fiance.
I remember he was very nice about it. She said something like: "Any woman would be lucky to be the wife of Chomely Finchel (sp?). And he said, "Well, you just may get me one day..after this family is raised to your satisfaction."
It was really a very sweet show. A little saccharine at times, maybe, but its' heart was in the right place and it had a very likable and talented cast.

I remember a more comic episode that played with the unspoken attraction between Nanny and the Professor.
Nanny was taking a college course on "The Art of Relationships" or something similar, taught by a guy who was a local ladies' man. (Nicknamed "Swinging Sam," of all things!) Of course, he was very attracted to Nanny and began dating her. On the night he tried to propose to her while she was sitting on a couch in the family living room,, he kept getting interrupted by the kids. Waldo (the family dog for those who have't seen the show) had a toy that had fallen under the couch, so Nanny asked the guy to get it out.
While the guy was on his knees trying to reach the toy (and talk to Nanny at the same time), the Professor drove up to the house with his date and saw them through the lighted window. His first impulse was to rush in and break it up, and he sort of fumbled to explain to his date "how hard it would be to find someone to replace Nanny", etc., when you could tell he was a little jealous of Nanny's date. The Professor's date was pretty nice about it all (and very pretty herself, as I recall!) and reminded him that he had to let Nanny live her own life, so he didn't do anything.
In the closing scene of the episode, the Professor and Nanny were talking by the couch about the boyfriend (who was now engaged to someone else) while the Professor was waiting for his date to pick him up. Waldo's toy again got stuck under the sofa, and this time the Professor got down on his knees while Nanny was sitting on the couch.
The Professor's girlfriend came up the walk to pick him up for their date and saw the Professor on his knees talking to Nanny. Based on the prior incident, she assumed the Professor was "proposing" to Nanny and got back in her car and drove off.
The Professor heard the car starting and driving off and realized it must be his date. He then recalled that he was on his knees, guessed what she must have been thinking, threw up his hands and laughed to himself.
Anyone else remember this episode?

Probably my favorite episode of the series glad a friend of mine had the foresight to tape the show when it ran on FX back in the mid 90's and then transfer it to DVD for old TV show junkies like me. (slightly cut, but way better than no episodes at all)
The episode was written by Jean Holloway, who also wrote a number of episodes for The Ghost And Mrs. Muir my all time favorite show.
As a matter of fact, a tall tale that Chomledy (sp) tells about being lost in the desert and when his camel was too tired to carry him, he carried the camel was lifted straight from a Ghost and Mrs, Muir episode called Double Trouble, in which Edward Mulhare (the ghost)had a double a handsome Irishman out to charm Mrs. Muir.
Nanny was a sweet show!

I'm a Cinemajunkie !
I have a memory like a steel trap for old tv shows.
Yes I remember that episode. Nany later offerd to tell the little girl a bed time story. Nany was standing next to the little girls bed.
The little girl said she would make up her own story and she did. Then she started to tell a story of her self and her family (In the third person) meeting the Nany. Being happy living with the Nanny, then in a sad way saying the Nanny was gone.
It was a sad espisode. Plus the little Girl was Actress Kim Richards.

I used to enjoy watching this show as a child. It was so long ago that I don't remember any specific episode. I do remember that there was a girl that lived in the neighborhood named Francine and she had a crush on Hal (the oldest boy).