ungeheuer
extremely
Used as an adverb to strongly intensify another adjective or a verb; extraordinarily, very.
Der Film war ungeheuer spannend bis zur letzten Minute.
The film was extremely exciting right up to the last minute.

enormous / extremely
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extremely
Used as an adverb to strongly intensify another adjective or a verb; extraordinarily, very.
Der Film war ungeheuer spannend bis zur letzten Minute.
The film was extremely exciting right up to the last minute.

The word comes from 'un-geheuer'. Something that is 'geheuer' is safe and trustworthy. So 'ungeheuer' originally meant 'eerie' or 'not safe', from which today's meanings 'immense' and 'extreme' developed.
In dem alten Schloss war ihm nicht ganz geheuer.
He did not feel entirely at ease in the old castle.
You can recognize the function of 'ungeheuer' from its position and form. As an adverb, it remains unchanged before another adjective (e.g., 'ungeheuer schnell' — extremely fast). As an adjective, it stands before a noun and is declined (e.g., 'ein ungeheurer Erfolg' — an enormous success).
Das Konzert war ein ungeheurer Erfolg.
The concert was an enormous success.
When 'ungeheuer' is used as an intensifier like 'sehr' (very), it often sounds somewhat more elevated or stronger. It gives the statement more weight than everyday intensifiers.
Die Berglandschaft war von ungeheuer erhabener Schönheit.
The mountain landscape was of extremely sublime beauty.
The original meaning 'monströs' (monstrous) or 'unheimlich' (eerie) is found today mainly in fairy tales and legends, or when something is described as frightening. The related noun is 'das Ungeheuer' (monster).
Aus der Tiefe des Sees stieg eine ungeheure Kreatur empor.
A monstrous creature rose up from the depths of the lake.
very
'sehr' and 'ungeheuer' can both strongly intensify something. 'sehr' is neutral and fits many normal statements. 'ungeheuer' as an intensifying word sounds clearly stronger and often a bit more dramatic or elevated. It fits well when something is extraordinarily large, strong, or impressive. You should not confuse it with the other meaning of 'ungeheuer', where something seems uncanny or frightening. In simple everyday sentences, 'sehr' usually sounds more natural.

Die neue Aufgabe ist ungeheuer wichtig für das ganze Team.
The new task is extremely important for the whole team.
Wir freuen uns sehr auf das Essen im Restaurant.
We are very much looking forward to the meal at the restaurant.