Dragon Ball, DB Z, DB GT/How much of a contributions did Old Kai play in Ultimate Gohan power up?
* Hybrid Saiyans [Like Gohan, Goten and Trunks] have potential to have higher ki levels than full Saiyans.
* During the Cell Games saga, Gohan reached SSJ2.
* During the Martial Arts Tournament [Babadi Saga], Gohan reached SSJ2, but Vegeta commented on that Teen Gohan SSJ2 was stronger.
* Goku's SSJ2 level is higher than Teen Gohan's SSJ2.
* Later Majin Vegeta SSJ2 has exactly the same power level as Goku's SSJ2. And Vegeta commented on that both [Vegeta and Goku] now is stronger than SSJ2 Teen Gohan.
* Gohan gets his ability unlock. Not only Old Kai raises Gohan's ki to the same level as SSJ2 Teen Gohan, Kai says he can extend his power "wayy past his limits".
Examples: Let's just use a scaling system out of 10.
10 = is the strongest
1 = is the weakest.
10 = Ultimate Gohan
9 = SSJ3 Goku and Kid Boo
8 = Fat Boo
7 = SSJ2 Goku and Majin Vegeta
6.5 = SSJ2 Teen Gohan
6 = SSJ2 Adult Gohan
Now we know for a fact that Goku's and Majin Vegeta SSJ2 had a higher power level compared to SSJ2 Teen Gohan. Plus Adult Gohan was said to be weaker than his younger self.
SSJ2 Goku & Majin Vegeta > SSJ2 Teen Gohan > SSJ2 Adult Gohan
After power up:
Ultimate Gohan >> SSJ3 Goku & Kid Boo >>>> SSJ2 Goku & Majin Vegeta.
Question: How much is Ultimate Gohan's power is "way past his own limits"?
Well, for starters Gohan was a teenager during the Buu Saga. Funimation altered Gohan's age way back in the Saiyan Saga because his original age, 4, was considered inappropriately low to be facing such life threatening situations. As such, he was changed to 5 and a half, and because of that, his age was consistently altered through the remaining sagas to accommodate that. In actuality, Gohan was only 9, biologically 10, when he fought Cell, and was only 16 during the events of the Buu Saga.
To answer your question though, Rou Kaioushin's power up, without going past the limits, wouldn't have just brought him up his Ssj2 battle power. It brings them beyond their maximum natural potential, and therefore stronger than what Gohan would have ever been able to become naturally. To use a different character, say Goku during the Buu Saga didn't have access to Super Saiya-jin 3. He was as strong as he was otherwise, just didn't have access to Ssj3. Rou Kaioushin's ability would have brought his strength up to what it was as a Super Saiya-jin 3 and beyond, even though in that scenario he couldn't reach that form.
With Gohan, it's the same way. His potential, being higher than that of his father's given that he is a hybrid, would have pushed him past Ssj3 Goku normally if Rou Kaioushin's ability just brought him to his maximum potential. However, in going beyond that, it pushed him even further past Ssj3 Goku than he would naturally have been able to.
Given information from the manga, I can give a rough estimate as to just where Gohan after the power up was, though without knowing what his true potential was normally I couldn't give you an estimation as to just how much Rou added.
Goku essentially establishes that Ssj Gotenks, before the Room of Spirit and Time training, will still be stronger than he (Goku) is as Super Saiya-jin 3. Even if you don't take this as solid fact, the fact that Goku believes wholeheartedly that Ssj Gotenks could defeat Fat Buu still establishes that Ssj Gotenks is above Ssj2 Ma-jin Vegeta (and therefore above Ssj2 Goku since the two were equal).
Ssj Gotenks > Ssj3 Goku
Ssj Gotenks > Ssj2 Goku
The Daizenshuu makes a statement about Gotenks' training in the Room of Spirit and Time that suggests that he, in his base form, managed to pass Ssj2 Vegeta in terms of power.
Base Gotenks (Post RoSaT) > Ssj2 Vegeta.
Given the Ssj multipliers of the Super Exciting Guide, this would put Ssj3 Gotenks 400x stronger than Ssj2 Vegeta (Ssj is 50x base, Ssj2 is 2x Ssj, and Ssj3 is 4x Ssj2), which in turn puts him 100x stronger than Ssj3 Goku.
Gohan, after his power up, was far stronger than Evil Buu (Super Buu), who was essentially equal to Ssj3 Gotenks in terms of power. However, after absorbing Gotenks, Buu became more powerful than Gohan, which if you assume Buu's absorption works like addition, would put Gohan somewhere within the range of 1.5x the strength of Ssj3 Gotenks, or about 150x stronger than Ssj3 Goku.
Not knowing just how much more potential Gohan had over his father, it's rough to say how much of that 150x was from Rou's power up, but I would safely say that most of it was from Rou, and Gohan's potential wasn't an insanely higher amount than Goku's.