So let's take a look at usage in context. First, the obvious, more straightforward meaning.
Ha scritto ai giornali, ha scritto al sindaco, ha addirittura scritto al Presidente della Repubblica.
He wrote to the papers, he wrote the mayor, he even wrote the President.
The word addirittura (which in the above case can easily be replaced by persino) expresses the idea that something or someone has gone beyond the bounds of expectations in their actions, speech, or opinion.
(Woman- angered) Sei un mostro!
(Man- nonchalant, ironic) Addirittura.
You're a monster!
That bad, huh?
Per nascondere l'identita' si e' addirittura fatto crescere la barba e ha tinto i capelli.
To conceal his identity he went so far as to grow a beard and die his hair.
E' cosi' arrabbiata per gli insulti della suocera che vuol farle causa.
Addirittura!
She's so mad about her mother-in-law's insults she wants to sue her.
No kidding!/You're kidding!/Really?!
Ha cercato lavoro dappertutto, e' andato addirittura in Estonia.
He looked for work everywhere, he went as far as Estonia.
Words like this should convince you that you can't learn a language just from the dictionary, because human languages are not strictly equivalent codes with a one-on-one correspondence. It will take time and practice and a sensitivity to context and nuance to get this right. E un bel giorno potresti essere addirittura piu' bravo di me (ma lo dubito). And one fine day you might be even better than I am (but I doubt it).
