It seems that you can convert ArcView .adf files to TIFF image format (.tif, .tiff) using the Total GIS Converter or similar tool. Probably used to export maps or some of their meta data as normal pictures so they can be viewed by everyone in any graphics viewer.
Convert ArcView ARC/INFO coverage data to Aldus Tagged Image File Format (TIFF) bitmap image.
We have found one software suitable in our database for this conversion.
Microsoft Windows software - convert adf to tiff on Windows
A converter that is able to transform various GIS file formats
Convert from adf file format
The table with program actions contains information about what each program is capable of doing with their files. This may be a good pointer for further searches for certain file conversions and help you find the converter you are looking for if our search result is insufficient for you. However, it is far from perfect and may sometimes show results which are not really usable because of the nature of the software handles the files. So a conversion that appears as possible, just because of matched actions, may in fact not be possible.