I had the privilege of an extended one on one with her aged about 18 or 19.

I had gone with my parents to the first night of a play "The Old Country" at the Haymarket theatre, starring Sir John Gielgud and Sir Ralph Richardson.
In the interval we went to the bar and my mother asked me to get her a glass of wine, however the barman said they had none. I was a bit disappointed as the very elegant lady standing next to me had a nearly full bottle. As I was about to convey the bad news to my mother the lady in question courteously said "what a shame - it's so disappointing when you can't get what you want. Please ask your mother if she would accept a glass from me." I smiled and she pour a glass which I took over to my mother along with my stepfather's whiskey.
Not wanting to be rude I then went back to thank the lady and we had a long conversation about the play, what I was hoping to do in life, all sorts. She was wonderfully attentive and a genuinely warm person. Anyway when the bell rang to return to the seats and I said goodbye and we went back to our seats.
When I got back my mother said "Did you get her autograph then?" I looked confused. "Who?" I enquired. "Liz Taylor, you just spent thirty minutes with her," my stepfather laughed. "You sods! Why didn't you tell me?" I asked. "Because she was so obviously enjoying the fact that you hadn't recognised her, and she could have a normal conversation for once," they replied. "We didn't want to spoil it for her."
So that was my full 30 minutes one on one talking alone with one of the greatest film-stars the UK has ever produced, and as usual with me, at the time, I was blissfully unaware of my good luck.
But then that's my life to a T. I do get fantastic good luck, but all too often I fail to recognise it!