User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-6768393-20130318025701/@comment-1605058-20130318123655

There is no largest ordinal definable in FO logic (because its succesor would be trivially definable too). What you probably meant is smallest number which isn't definable. It may not be ordinal which we seek, but regardless of it we can define larger ordinals, fundamenta sequences for them and thus faster growing functions. I think