Because my grandmother was insanely cheap, we had this thing where she would call and hang up after one ring, and then we would call her back. Then the phone company got priority ringing, so we just wouldn't pick up when she called.
Anyway, your thing about double-ringing reminded me.
"Leave a missed call" was so popular in South Africa that the networks implemented "Send a call me", where you send a GSM code with the target person's phone number and the network sends that person an sms message from your number with the text "please call me", your number, and an advertisement.
Normally sending messages attracts a charge per SMS but a "please call me" is free, advertiser supported and limited in how many one can send per day.
Also known in the UK as a "one-ringer" or "missed call" (as in - "I'm running low on credit, I'll just missed call you when I'm there?"). Amusingly enough, my brother, who had eked out 10p of credit for several weeks this way, once got quite annoyed when I was a little too prompt to answer the phone...
Anyway, your thing about double-ringing reminded me.