As I understand it the word order isn’t ideal, it could even said to be such as to misrepresent the true idea; I think that what is meant must be put as either of the four following options;

• Je sais ce dont j’ai besoin / toujours d’aller encore plus loin
• Je sais ce dont j’ai besoin / toujours d’aller plus loin encore
• Je sais ce dont j’ai besoin /d’aller toujours encore plus loin
• Je sais ce dont j’ai besoin /d’aller toujours plus loin encore

Keeping to the initial order there is first the concept “going further without ceasing” (toujours plus loin), “without ceasing” being the rendering of “toujours”; then, adding the last modifier (encore toujours plus loin), we get that this action of going further without ceasing has to be performed again (encore); there is no notion of how many times. This is not the intended idea in my opinion; I believe that the idea, quite a banal one nowadays, should I say, is that of “always going still further” ( always to go still further).

Some paraphrasing might be useful; the idea is that the ideal in the way of making one’s existence worthwhile is never to cease after each inch of ground gained against adversity or gained by the plain exercise of one’s powers, to go one inch further, in other words to go always (toujours) one step further (plus loin) after the last step (encore).

The translation is then this;

I know what I need / to go always still further

That of “Je sais ce dont j’ai besoin / d’aller encore toujours plus loin”
is not “I know what I need / to go even further” but “I know what I need / to go again further forever”, which is rather nonsensical, wherefrom my opinion the original idea has been misworded and the changes made.

It is more common and standard to say encore et toujours. It is quite possible the author simply skipped the conjunction for the lyrics to better match the music.

Both adverbs have a a slightly different meaning but when combined that way, share the repetition / persistence idea so the expression is pleonastic.

• encore: one more time, once again
• toujours: always, every time

The core part is toujours plus loin meaning “ever further” (not “even further”) while encore is there to anchor the expression to the present:

I know what I need / to go, once again, ever further

TLFi:

Un poulet, encore du poulet, toujours du poulet, nom de Dieu ! J’en ai assez, moi, de ton poulet (Maupass., Contes et nouv., t. 1, Alex., 1889, p. 758).
…
c) Lorsque, dans la lang. fam., ces adv. se trouvent combinés, leur alliance, rare et pléonastique, équivaut à toujours au sens de la persistance, encore se bornant à suggérer la relation avec le moment du procès. D’autre part, je commençais à mépriser Yves, mais je l’adorais encore toujours (Drieu La Roch., Rêv. bourg., 1939, p. 288).