You mention in the comments that user studies have shown this to be an important tool, but I'm not sure that it necessarily means that the x-scales need to be different and on top of each other. It is after all the same axis (days), just in different resolutions. It sounds to me that the right way to display this is to show all three lines, use a monthly or weekly basic scale for the X axis, and to let users zoom in and out to get to a daily/quarterly/yearly resolution, e.g. like Google does. And then if you get to the daily scale but one of the datasets only offers yearly data points, it will "stretch" between the two closest points.
Usually when users request data to be shown together it means that they're looking for correlations or other trends in the data across datasets. In this case it doesn't seem to make sense - a rising trend on a specific Sunday-Monday-Tuesday will seem to correlate with a rising trend on a specific January-February-March. I don't think that data on different orders of magnitude of the same dimension should be displayed together on the same axis like this, especially if it means that the same data point can appear in several places on the chart (separate locations for a specific date on the daily, quarterly and yearly scales). To me this seems not only unclear but actively misleading the users.
I would try to investigate the need further.