The Jargon File (version 4.4.7, 29 Dec 2003):

annoyware
 n.

    A type of shareware that frequently disrupts normal program operation to
    display requests for payment to the author in return for the ability to
    disable the request messages. (Also called nagware) The requests generally
    require user action to acknowledge the message before normal operation is
    resumed and are often tied to the most frequently used features of the
    software. See also careware, charityware, crippleware, freeware, 
    FRS, guiltware, postcardware, and -ware; compare payware.


The Free On-line Dictionary of Computing (30 December 2018):

annoyware

    Shareware that reminds you frequently that you
   are using an unregistered copy.

   (1998-04-29)