As Russ says, you can set the minimum and maximum zoom scales when you download the map. Just be sure to set the Work Area to the area you need, then set the Map Detail to the maximum zoom that you will ever want to use, then Download.
For example, if you choose a Work Area with a scale of 1:50,000, the default maximum zoom is 1:25,000. However, after setting the Work Area, you can set Map Detail to allow zooming down to 1:1200 (depending on basemap) if you want.
It will increase the size of the download when you do that so keep an eye on the estimated size.
Is it possible to zoom to greater than 1:1200, to say 1:200 when using a custom basemap?
If your basemap has that tiling level available you will be able to zoom to it.
Russell, I know what you're saying, but the maximum scale is 1:128.49 when you use the tile package dialogue in ArcMap. Is there a work around to get it to 21 levels, rather than 20?
You will have to use your own tiling scheme instead of using the ArcGIS tiling scheme to get down to that level.