From AmtWiki

The squire of one's former squire who is now a knight. Only knights have grand-squires.

A related term is grand-knight.

Amtgard has several belted lines famous for their length. Sir Auromax's consecutive Sword Knights come to mind. Auromax squired Lief who squired Sparhawk who in turn squired Ice. All four were Sword Knights. The tree goes even deeper since Auromax himself was squired to Thorn Crossbearer but Thorn wasn't a Sword Knight. Since knights frequently have multiple squires this leads to an entire 'knightly family'.