What is an adjective for the word continually?

There isn't a single perfect adjective for "continually". However, depending on the context, you could use:

General:

* Constant

* Persistent

* Unceasing

* Incessant

* Perpetual

* Enduring

* Continuous

* Relentless

More specific:

* Everlasting (for something that never ends)

* Chronic (for something that happens frequently or for a long time)

* Habitual (for something that is done repeatedly)

* Recurring (for something that happens again and again)

The best choice will depend on the specific situation and the nuance you want to convey.