As a MediaWiki system administrator, I want to be continuously harassed about my MediaWiki installation being in need of upgrades, so that I don't forget.

Nowadays CMS software alerts you pretty much anywhere in the "edit" interface (see Wordpress, Joomla, IIRC Drupal). In our case the best chance is probably to intercept some page that users/editors see, like Special:Version, so that they nudge the sysadmins (who are typically lazy, careless and uninterested in the wiki).

This would be trivial to do if vanilla MediaWiki core updated system messages automatically: we'd just have to add a message on Special:Version (or other special page chosen). Cf. «If we had this, we could also use some special system messages, or qqq messages, to spread crucial information or docs (see also bugzilla:41578). --Nemo 21:28, 10 November 2012 (UTC)».

According to MzMCBride, however, https://www.mediawiki.org/wiki/Architecture_meetings/RFC_review_2014-12-03 may have decided that

adding
a button that when pressed checks for updates to MediaWiki core and its
extensions and reports the results

depends on https://www.mediawiki.org/wiki/Requests_for_comment/Opt-in_site_registration_during_installation (does that have a task?).