No, you are not doing something wrong.
Actually, a few minutes print time for a highly detailed vector basemap with maybe a few dozen vector layers, is not bad at all! The amount of processing needed to convert this to something printable, is quite considerable.
The particular behaviour you are seeing, with repeated cycles of the same layer names appearing in a kind of loop, is actually normal and quite typical of exporting or printing a layout in ArcMap. It has to do with the display pipeline of ArcMap, and this is one of the areas where the new ArcGIS Pro significantly differs from ArcMap.
This behaviour is triggered and determined by having vector layers with transparency set over raster layers, like WM(T)S basemaps. When you export your map, ArcMap may need to rasterize some of the vector layers depending on how you stacked the layers in the TOC. This process seems to be done in small chunks / tiles across the printed area, hence the repeated redrawing of the same layers.
The amount of times the layers are re-drawn, seems directly related to print resolution in a quadratic way. If you set a very high print resolution (e.g. on exporting of a PDF you set a 1200 or 2400 ppi resolution), the printing or exporting time may thus become prohibitive compared to printing in more "normal" resolutions of 300-600 ppi.
E.g., if you set a 300 ppi resolution, and it takes 2 minutes to export, then the equivalent print on 2400 ppi may take 2 min x (2400/300) x (2400/300) = 2 min x 8 x 8 = 128(!) minutes... so more than 2 hours!
So try reducing the printing resolution to say 300 or 450 ppi any time you use transparency in your maps.
NOTE: this issue does not occur in 100% vector maps. You can safely export to 1200 or 2400 ppi without significant delays (I have done that hundreds of times with complex A0 size layouts). And yes, a higher resolution will improve the quality of exported vector (PDF) output, especially things like text halos will render much better at higher ppi output. You may not notice the difference when viewing the PDF in Adobe Reader at 100 % zoom, you will notice the difference in quality when zooming in to maybe 1600 % in Adobe Reader. It may actually sound surprising that changing the ppi settings, affects 100 % vector output at all, but this is caused by the fact that stuff like halos, aren't stored as true vector curves like fonts / glyphs are, but instead stored as small straight line segments in the output. The higher the ppi, the more detailed and more faithfull the resulting "pseudo-curve" will be.
Lastly, I do strongly recommend you to first export your data driven page layouts to PDF, and then print the PDFs instead of printing straight from ArcMap. This way, you only need to take the hit of the ArcGIS exporting time once, and can re-print the layout any time you desire. Additionally, you have full control over the exported ppi resolution by setting it upon export to PDF.