Arjun,
Dhananjaya
Gudaakesh - Who has won sleep.
Jishnu
Keerti
Kuntee-Nandan - Who is Kuntee's son.
Kuntee-Putra - Who is Kuntee's son.
Paandu-Nandan - Who is Paandu's son.
Paandu-Putra - Who is Paandu's son.
Paarth
Parantap - The subduer of enemy.
Phaalgun
Savyashachee
Shwetvaahan
Vibhatsu
Vijaya

(2) In Geetaa Arjun has been addressed by Krishn by 13 names -
(1) Arjun,  (2) Paandav,  (3) Dhananjaya,  (4) Gudaakesh,  (5) Paarth,  (6) Kaunteya,  (7) Parantap,  (8) Bhaarat,  (9) Kurunandan,  (10) Bharatarshabh,  (11) Kurusattam,  (12) Kurupraveer, and  (13) Bharat Shreshth.

In Mahaabhaarat, when Arjun was in A-Gyaatvaas in Viraat's city, and goes to fight with Kaurav with Uttar, he collects his weapons from the Shamee tree where Paandav kept them in hiding. Uttar wonders that how an eunuch can use Arjun's weapons, then Arjun reveals his identity to him, still he doubts, and asks Arjun's 10 names to remove his doubts. Arjn tells him the following 10 names to him --
(1) Arjun,  (2) Phaalgun  (3) Jishnu,  (4) Kireet,  (5) Shwetvaahan,  (6) Vibhatsu,  (7) Vijaya,  (8) Paarth,  (9) Savyashachee,  and (10) Dhananjaya.

Then he explains why he is called by these names -
I was called Dhanajaya when I conquered all the kings at the time of Raajsooya Yagya and collected wealth from all of them. I always fight till then end and I always win, that is why I am called Vijaya. My horses which were given to me by Agni Dev are white, that is why I am called Shwetvaahan. My father Indra gave me a beautiful crown when I was with him, that is why I am called Kireet. I have never fought by unfair means in any battle that is why I am called Veebhatsu. I never frighten my enemies by meanness, I can use my both hands when I shoot my arrows, that is why I am called Savyashachee. My complexion is unique like Arjun tree, and my name is stainless, that is why I am named Arjun. I was born on the slopes of Himvaan in a place called Satsring on a day when the star Uttaraa Phaalgunee was in ascent, that is why my name is Phaalgun. I am called Jishnu because I am terrible when I am angry. My mother's name is Prithaa, so I am also called Paarth.