Like a doorbell that chimes twice when the button is pushed once? Do you already have the chime? Are you thinking of an electronic chime (speaker).
The minimum component circuit would use an 8-pin microcontroller. However that requires programming.
Ken - thanks for your questions. I do not have a chime and more specifically I want an electronics chime that when a button is pressed it would chime 11 times with it's output being a small speaker. I am a Electronics Tech with a Motorola dealer for the past 20 years and prior to that I spent 20 years in Electronics in the Air Force. Although I can follow schematics, troubleshot and solder, my skill set in design sadly lacks.
Sounds like you need two multivibrators (one to make the note, the other to make the pulse duration/period), and possibly a counter to count pulses to shut it off, or possibly a period timer that shuts it off after a fixed time.
Attached are some circuits that I've used to make SONALERT's into chimes, like the headlight chime in your car. I think that you could use the 555 circuit, but use a LM556. The second half could be a monostable that would set the number of chimes.
This is what I had in mind (not bench tested or simulated). No Idea what your trigger is to start the chiming.
UAI is a power-up long-duration monostable and UA2 is a short-duration pulse astable.