impudence / outrage
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outrage
A single action or statement that is perceived as very brazen, disrespectful, or improper.
Nach der langen Wartezeit ohne Erklärung war seine knappe Antwort eine Unverschämtheit.
After the long wait without an explanation, his curt answer was an outrage.
The word consists of three parts: the prefix 'un-' (not), the adjective 'verschämt' (shy, reserved), and the ending '-heit', which denotes a quality. An 'Unverschämtheit' (impudence/outrage) is therefore literally the behavior of someone who is not ashamed of acting brazenly.
Seine Forderung war eine absolute Unverschämtheit.
His demand was an absolute outrage.
You can distinguish between the quality and a single action. If you speak of 'seiner/ihrer Unverschämtheit' (his/her impudence) in an indefinite sense, the general quality is meant. If you speak of 'einer Unverschämtheit' (an outrage), you mean a specific, countable action or statement.
Seine Unverschämtheit kannte keine Grenzen.
His impudence knew no bounds.
As an expression of outrage, the exclamations 'Das ist eine Unverschämtheit!' (That is an outrage!) or 'Was für eine Unverschämtheit!' (What an outrage!) are often used. With these, you judge a situation or action as extremely disrespectful.
Dass er einfach gegangen ist, ohne zu bezahlen? Das ist eine Unverschämtheit!
That he simply left without paying? That is an outrage!
When the quality is meant, 'Unverschämtheit' (impudence) is often used with a genitive or a possessive pronoun to show to whom this quality is being attributed.
Die Unverschämtheit dieses Verkäufers hat mich schockiert.
The impudence of this salesperson shocked me.
hammer
'der Hammer' and 'die Unverschämtheit' can both be used when you are outraged about something very brazen or disrespectful. 'der Hammer' is a very colloquial exclamation and can also be meant positively, depending on the situation. 'die Unverschämtheit' is clearly negative and directly names disrespectful behavior or a brazen action. If you want to say clearly that something is not okay and is disrespectful, 'Unverschämtheit' is more precise. 'Hammer' shows the surprise or outrage more strongly.
Lea findet es eine Unverschämtheit, dass niemand sich für die Hilfe bedankt.
Lea thinks it is an outrage that no one thanks her for the help.
Dass der Kunde nach Ladenschluss noch eine Beratung verlangt, ist wirklich der Hammer.
The fact that the customer still demands advice after closing time is really outrageous.