QGIS crashes when changing projection to South Pole Lambert Azimuthal Equal Area

To summarize the answer that came out in the comments above...

The basemaps you're using (OSM and ESRI base maps) are not compatible with a polar projection. Like most basemaps, these are served in the Web Pseudo-Mercator projection (EPSG:3857). That projection doesn't extend to the poles. The valid extent of that projection is: -180.00, -85.06, 180.00, 85.06, which means it covers the entire 360 degrees along the east-west axis, but it only extends as far north as 85.06 degrees, and as far south as -85.06 degrees.

You need a special south pole basemap. Fortunately, these exist. Unfortunately, they don't seem to exist inside the Quick Map Services plugin, so you will have to learn to load a WMS layer through the Data Source Manager, which can be quite tricky. Here's a tutorial. The basemaps that come with the QMS plugin are loaded in XYZ tile format, so if you find a basemap in tiled format you can look at the settings of one of the QMS basemaps you already loaded, and use them as a template to help you get the settings correct on the other layer. See here for detailed steps.

With a quick google search for "wms antarctic basemap" I found the following basemaps:

• https://hub.arcgis.com/datasets/d13b9d10219e4429974e48368b64e41f
• https://gis.ngdc.noaa.gov/arcgis/rest/services/antarctic/antarctic_basemap/MapServer

WMS stands for "web mapping service," which is a format that many basemaps come in. They also come as REST services and XYZ tiles, so you might find different resources using those search terms.

I also found a few questions here on GIS StackExchange about polar basemaps that have some really promising-looking answers.

• Choosing polar projection and tiling scheme for web mapping of polar regions?
• Tiled Web Map; what tile provider offer antarctic map tiles?

It has nothing to do with base maps. QGIS does not work well with the South Pole Lambert Azimuthal Equal Area projection, regardless of the basemap used. I hope they solve it because you cannot work on projects with the Antarctic area as a study area, for example.

Type world to the coordinates field at the bottom of the QGIS window: it will load a vector shapefile showing countries. This basemap should reproject, see:

For me, using QGIS 3.20 on Win10, it even works with an OpenStreetMap basemap: