Fractal Zoomer is an comprehensive, portable and open-source application which makes it easy to explore the spectacular world of fractal art.
If you've not come across fractals before, they're mathematical sets with complex patterns which are the same (or almost the same) at different scales. You don't have to worry about the detail, though. The program opens with the standard Mandelbrot set, and all you have to do is repeatedly click at any point which seems interesting to zoom in.
If you're tired of Mandelbrot imagery then Fractal Zoomer offers plenty of alternatives. Click Options > Fractal Functions, choose an alternative starting point, and you can zoom in and move around as usual.
Find a view you like, and it can be tweaked in various ways. You might rotate it, for instance (Options > Rotate), apply a Sharpen filter (Filters > Sharpness), even repeatedly cycle its colours for a ripple effect (Tools > Colour Cycling).
There are plenty of other options available. If you don't understand the core math then most won't mean much (Options > Planes > 1/ (mu - 2), for instance), but don't let that put you off. They'll just give you a different view of the current image: try a few, see what happens.
If you locate a particularly interesting area of the fractal world then clicking File > Save As will save your current position for easy recall later.
Or, if you just want to save the current image, click File > Save Image As to record it for posterity.