In DBZ: Battle of Gods, Goku became Super Saiyan God and he makes Beerus use upto 70% of his energy. So, is it possible for Goku to defeat Beerus in his Super Saiyan God Super Saiyan form?
There is no clear answer. But according to the Wikia article on SSGSS:
The form exceedingly surpasses its predecessor, Super Saiyan God, and all of the other Saiyan forms.
This source also mentions that SSGSS will appear in Dragon Ball Super, so there might be a chance, but we can't be sure unless we watch Dragon Ball Super..
As he is right now Goku is weaker than Bearus (it was said in the universe 6-7 tournament that maybe the 2 of them together could give Beerus trouble).
The 70% stuff was changed. In the anime Beerus just says he didn't use his full potential (otherwise Goku would already be several times stronger than Beerus).
So yeah, he may reach him by the end of the anime
I think Vegeta could defeat Beerus. After all, he did unlock Super Saiyan Blue evolution which cuts through the energy of destruction, something Goku hasn't been able to do yet.
Goku could take on Beerus in Super Saiyan Blue Kaio-ken times 20 since Toriyama did confirm SSJBK and SSJB evolution to be at the same level. Super Saiyan God? Maybe if Beerus was already at 100 of his power and ran out of stamina and stuff and Goku uses it at the right time and beats him somehow. Or maybe even go SSH God Kaio-ken to save stamina since Super Saiyan God Blue drains more ki than the regular Super Saiyan God form.
Well now we know the answer is no. In the tournament between universe 6 against universe 7, Hit was at the level of Goku Super Saiyan Blue Kaioken x10 (Super Saiyan God Super Saiyan Kaioken x10) . And Hit was threatened by Champa who is in a similar level than Beerus. Goku wanted to help Hit, because he knew Hit likely wouldnt be able to defeat Champa alone (and probably he wouldnt be able to defeat him at all). So if Goku super saiyan blue Kaioken multiplied 10 times might not even be at the level of Beerus-Champa, even less super saiyan blue with no multiplier. Plus in the manga they say Vegito (Goku and Vegeta fusion) seems to be in the level of Beerus. And Fusion adds the power of both fighters and multiplies it "tens of times" (according to Vados). So most likely Super Saiyan Blue is "tens of times" weaker than Beerus. Ultra Instinct Goku might be in the level or above Beerus, since Beerus didnt master Ultra Instinct (and Goku sort of did, he could use his full power/abilities but for a while)