(Future) Trunks Can Reach Super Saiyan 2
The following statement was true up until 2016: “Trunks is never shown to be able to reach Super Saiyan 2.” Trunks is shown as able to reach Super Saiyan, and both stages past that within the same transformation (that is to say, Grade II and Grade III)… but not Super Saiyan 2.
The Budokai video games added to the confusion. The first game did a good job of sticking with the traditional Japanese naming conventions (“Super Saiyan Trunks” and “Super Trunks” for Grades I and II, respectively). The following year, Budokai 2 changed this with a “Super Saiyan 2” transformation capsule for Trunks. Unfortunately, this is not even a translation error; it is actually (incorrectly) called the Super Saiyan 2 capsule for Trunks in Japan, as well!
These two in-between stages would each be considered, as Goku puts it, ichi-dankai no henshin (“transformations of the first stage”). However, these two stages are thought of by Vegeta and Trunks as Sūpā Saiya-jin ni-dankai and Sūpā Saiya-jin san-dankai (“Grade II / Grade III of Super Saiyan”, with ni and san meaning “two” and “three”, respectively).
Official guide books also backed this up, time and time again stating in charts that Trunks never achieves anything beyond the three stages of the first Super Saiyan transformation.
Before we make our way to Dragon Ball Super, we should address some of the other, modern video games. Trunks actually received a Super Saiyan 3 transformation in the card-based arcade game Dragon Ball Heroes, which was then extended to his Dragon Ball GT incarnation when that update hit in early 2012:
The answer to this Super Saiyan 2 question for Trunks was somewhat-definitively updated and explained during Episode 49 of the Dragon Ball Super TV series in June 2016. Trunks escapes the wrath of “Goku Black” using his time machine, and winds up in the care of his friends in the past. As he sizes Goku up once more (similar to the first time Trunks appeared in the series), Trunks powers up through Super Saiyan. As electricity sparks around him, Goku specifically comments that Trunks is using the Super Saiyan 2 transformation.
Trunks and his Super Saiyan 2 transformation is further explored in chapter 16 of the Dragon Ball Super manga, as well as in the “Trunks: The Warrior of Hope” paid downloadable content for the video game Dragon Ball Z: Kakarot, which directly adapts said chapter.
Canonical discussions aside, Trunks has now in some form been shown to reach Super Saiyan 2!