Which algae grow fastest under optimal conditions?
The answer to your question depends on many variables. To narrow down the list of possible species it might be helpful to list under which conditions you intend to grow this algae and whether you want to give an additional carbon source such as glucose or acetate.

To give you an idea why this is important: the fastest growing alga I have ever worked with is Chlorella sorokiniana with double timings in the range of two hours. However this was in acetate containing medium and it would not have relied a lot on photosynthesis to grow nor would this make a lot of sense in open pond cultivation systems.

In terms of photoautotrophic algae, we have seen the most consistent quick growth (OD) in our system (Algem photobioreactor) for Dunaliella species. But this is based more on experience than rigorously testing a lot of species. In general there also seems to be a correlation between size and growth speed. Some cyanobacteria have a growth rate of around two hours Article