Preliminary remark: I created the maps here using EPSG:3857 (Web Mercator) for demonstration purpose only and because projection is not the issue of your question. For many tasks, however, this projection is not ideal and you should think about whicht other projection to use.
To create "Powers of ten"-like maps, you can use the atlas-fuction of the print layout: see Creating Detailed Maps with the Atlas Tool and documentation. The basic idea is to create a polygon layer (so called Coverage layer) containing a separate polygon for each of the different scales/extents you'd like to have and then create a separate map for each of these polygons.
Create polygons for the coverage layer. Create a point-layer with a single point where the map should be centered. For demonstration purpose, I used a point in Andorra.
On the point layer, run Menu Processing / Toolbox / Geometry by expression with the following expression (cf. screenshot 1 below). It creates squares centered on the point with side-length of 10, 100, 1000 and 10000 km. You can change these values, adding others or deleting some in line 3. Alternatively, to automatically generate a series with power of 10s, you could also use array_foreach (generate_series (1,7), 10^@element) for more than just 4 power of 10 values - here from 10 (10 to the power of 1) to 10000000 (power of 7):
collect_geometries(
array_foreach (
array(10,100,1000,10000),
make_regular_polygon(
$geometry,
project ($geometry, @element*1000/2,0),
4,
1
)))
On the output of step 2, run Menu Vector / Geoemtry tools / Multipart to singleparts to get a separate feature for each square. Use the resulting layer of this step as coverage layer in step 4.
Create a new print layout (see documentation), add a map (see screenshot 2: no. 1). On the atlas tab (no. 2), select the coverage layer. Go to the tab Element properties (no. 3) and check the box next to Controlled by Atlas.
You're done! Click Preview atlas (no. 4) to click through the different maps in the different scales you defined. Use Menu Atlas / Export Atlas as [Images… / SVG/PDF] to save a map for each scale defined in the selected file format.
Screenshot 1: showing how the expressions generates the squares for the coverage layer (red outlined frame), here using Geometry generator for demonstration purpose - however, you should use Geometry by expression instead as described above:
Screenshot 2: Making the atlas settings in the print layout - here showing page 3 of the atlas, representing the 1000 km square:
Screenshot 3 to 6: Output with the coverage layer (squares, red frames), based on a OpenTopoMap basemap: 10 km square:
100 km square:
1000 km square:
10000 km square:
