Here's the obvious and straightforward way, which has the major disadvantage of silently not updating if the manager changes and perhaps making you call the wrong number at an inopportune time:

Jane's manager's phone number is {{ChrisSmith!!phone}}.

Now for trying to make it grab the manager's name from the manager field on the fly. You may have been tempted to try:

{{ {{!!manager}}!!phone }}

…but unfortunately this does not work; TiddlyWiki doesn't parse wikitext within <<>> or {{}}, so variable and field transclusions cannot be nested.

Instead, you can use the $transclude widget and pass the transclusion of the manager's name as the value of the $tiddler parameter:

<$transclude $tiddler={{!!manager}} $field="phone"/>

It's possible to create additional levels of indirection (e.g., to get Jane's manager's manager), but to do it elegantly, we will need to know how to store the results of filters to a variable, which will be in an upcoming section.