The the popular meme shows, the flag of Norway is the “mother of flags”.
It doesn’t, however, contain itself (unless you don’t care about the relative widths of the white and blue parts to the whole). In other words, it is not self-similar.
A less-popular meme points out one national flag that has this property: the Czech Republic’s.
But there are more examples. For one thing, any bicolor flag has this property:
(The flags of Monaco and Indonesia, basically the same design, also contain copies of each other in themselves.)
Seychelles’ flag, though far more complex (five colors!), is also self-similar.
That’s it for national flags. Looking farther afield, there are many other bicolors and a few other tricolors of the Czech type. Single-color flags, like that of Qaddafi’s Libya, are self-similar, of course — uninteresting so.
Here’s a self-similar flag of four colors:
And, finally, there are the many heraldic gyronny flags. Here are two examples:
Are there other kinds of self-similar flags not covered here?
Portland Flag Association 
The flag of Ecuador literally contains a depiction of itself flown from a steamship going down a river.
The national flag of Benin also contains infinitely many smaller copies of itself!