What is the awakening limit for generic 1* summons?

In Final Fantasy: Brave Exvius; I got a couple of 1* summons (an archer and a black mage) to max level (15) and awakened them to become 2* (max level of 30). Once they reach L30, will I be able to continue the process, and if so how many times can this repeat? I certainly don't expect them to reach 6*, but find it interesting that they gain more abilities as a result of awakening.
Best Answer
Besides the list mentioned in Ben's answer, units that can level up also show extra stars in the rank in a different color (faded out to indicate the actual rank is not yet maxed). So looking at this indicator ingame will quickly show the current & max star ranks available for the unit. None of the summons that are initially 1* can level up to max rank.
The other thing to take note of, is there are some units which may awaken to a lower rank currently but the developer may add new awakenings in future (or might already have done so in the Japanese version). An example is Vivi who is limited to 4* for the global version but has 5* and 6* awakenings in JP version.
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