If someone forgot to count Sefirat HaOmer at night and therefore counted during the following day without a beracha (as per the halacha OC 489), and then forgot to count Sefirat HaOmer again the following night (and so counts without a beracha again the following day) may they resume counting with a beracha on the third night? Is there some maximum number of consecutive nights which may בדיעבד be missed after which one may no longer make a beracha?
As explained in the answers to Is there an obligation to Count the Omer without a Bracha? we see that
The Terumat HaDeshen (#37) writes that if you aren't sure if you missed a day, then you don't need to worry about the opinion of the Behag, since it's only a doubt about a minority opinion and we don't need to be that worried about Berakha Levatalla. This is the ruling of the Shulchan Arukh (OC 589:8). Counting one day during the daytime (or any other doubtful counting) is no worse than being unsure if one counted at all (Levush, etc., ibid.) and one should proceed to further days as if they are for sure obligated to count.
Since each time one has that problem, it is no worse a situation (because one has recovered correctly), the one should count the remainder with a bracha