
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.

enormous
Used as an adjective to describe an enormous, immense size or an extreme extent.
Er hat ein ungeheures Vermögen durch kluge Investitionen angehäuft.
He accumulated an enormous fortune through smart investments.

monstrous
Frightening, eerie, or monstrous in appearance.
In den alten Sagen ist oft von einem ungeheuren Seeungeheuer die Rede.
In old legends, there is often talk of a monstrous sea monster.
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.
incredible
Unglaublich and ungeheuer can both describe an extreme degree. Unglaublich usually also indicates that the extent is surprising or hard to believe; ungeheuer focuses more strongly on the enormous size itself.

Der Wiederaufbau des Bahnhofs erforderte eine ungeheure Summe für Material und Arbeit.
Rebuilding the train station required an enormous amount of money for materials and labor.
Der Wiederaufbau des Bahnhofs kostete eine unglaubliche Summe, die niemand erwartet hatte.
Rebuilding the train station cost an incredible amount, which no one had expected.