Often in old maps you see where the cartographer has "cheated" by allowing part of the map to extend over the frame. You can see an example at the top (or the bottom of the inset) of this map:
Is there any way to do this in QGIS print composer?
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There is a way to do so, not very elegant, but useful.
In your print composer, add another map and choose the same scale as your main map. Move the second map to fit it in the correct position:
Then, put it over the main map in the "items" panel to see it over everything:
Here you can see the final result (I've done it in a couple of minutes just to show you, so it is not polished). It is a little tricky to put everything correctly. But once you get used to it, it is quite useful and easy to handle.
As proposed by Ed Rollason, I add my comment to the answer:
If you don't want your "overlapped map" to be rectangular, and wants to style it something similar to your example (where the extra space out of the border has the same shape as the continent), you can use svg shapes, as seen here:
I was wondering how to do this more elegantly, and using part of my previous answer (svg shapes), i've tried another solution that may be more suitable for this issue. Here it goes:
First, draw a svg shape that has a rectangular part (the one that is going to be the frame) and add some polygonal parts where you want your map to overlap your frame. You have to be careful and methodical, because the size of your svg must be proportional to your composer map (at least the distance relations: how far from the edge of the rectangle is the "overlapped part"?).
You can create this using any svg-creation online tool (tip: copy the part of the map you want to represent and draw your svg over it) or using QGIS pre-existing shapes (as Victor pointed out):
You can then style the borders, so you get a more stylish frame (add a legend that fits in the black space, draw a classic coordinate box, etc.).