This fractal was made by repeatedly zooming in on ever smaller versions of a self similar feature. Each time another zoom point was picked it added another rotated square to the fractal.

Here is a link to an image:

The parameter file used to make the image:

Do_It_Again { ; Exported from Fracton.

 ; time = 45 sec at 640 x 480 with no aa

 ; on a Mac Pro 2 x 2.8 Ghz Quad-Core Xeon

 ; by Mike Frazier, public domain

 reset=2004 type=mandel passes=1 float=y

 center-mag=-0.637200981531797321929241216142613588\

 79610418726/-0.49499504327751979843855327780682656\

 575746254617/3.921568676470588e+24/1/43/0

 params=0/0/0/0/0/0/0/0/0/0 maxiter=4000 inside=0

 proximity=0

 colors=000y_TxZSuYRsXQpVOlSNhQLcNJZKGUHEOEBIA8C756\

 32000411932E54J75N87RA8VB9ZDAaEBdFCgGDiHDjHElIElIE\

 mIFlIElIEjHEiHDgGDdFCaEBZDAVB9RA8N87J75E5493241100\

 0310721A32E42H53K64N74Q85S95VA6XB6YB6ZB7_C7`C7`C7`\

 C7_C7ZB7YB6XB6VA6S95Q85N74K64H53E42A32721310421942\

 D73I94MB5QD6UF7XH8_J9bKAdMBfNBhNBiOCjOCjPCjOCiOChN\

 BfNBdMBbKA_J9XH8UF7QD6MB5I94D739424210002414927E39\

 J4BN5DR6FV7HZ8Ja9KdAMgBNiBNjBOlCOlCPmCOlCOlCNjBNiB\

 MgBKdAJa9HZ8FV7DR6BN59J47E349224100022644C77I99OBB\

 UDDZFFcHHhJJlKKpMMsNNuNNxOOyOOzPPzOOzOOyNNxNNuMMsK\

 KpJJlHHhFFcDDZBBU99O77I44C226000641C92IE3OJ4UN5ZR6\

 cV7hZ8la9pdAsgBuiBxjBylCzlCzmCzlCylCxjBuiBsgBpdAla\

 9hZ8cV7ZR6UN5OJ4IE3C92641000413937D5AI6EM8HQ9KUBNX\

 CQ_DSbEVdFXfGYhGZiH_jH`jH`jH`iH_hGZfGYdFXbEV_DSXCQ\

 UBNQ9KM8HI6ED5A937413000632C75IA8OEBUHEZKGcNJhQLlS\

 NpVOsXQuYRxZSy_Tz`Tz`Uz`T }