In one-way ANOVA F=MStreatment/MSerror. F will be large if MStreatment is large compared to MSerror. MSerror is the within treatment mean square and MStreatment is the between treatment mean square. If each treatment level has very little variability in response and the different treatment levels have large variability one another, the result will be a large F. It is difficult to peg down the F-value. Yet I have not seen reason why the F cannot be so large.

You could log transform the data. However, doing so because you have a large F value is inappropriate.

Are all the assumptions for ANOVA satisfied? If so, then I don't see a problem and would agree with Ette's response.

Different time intervals suggest repeated measures design might be required.

It all depends on the data. The largest F I ever reported was over 11,000. It was on pHs in our treatments (which we were controlling) we had an effect size of 0.5, SD of 0.04 and an N of 200 per treatment. I wasn't even going to run a ANOVA but one of the reviewers insisted on it! See the first sentence in the results if you're curious (Long, W.C., Swiney, K.M., Harris, C., Page, H.N., and Foy, R.J. 2013. Effects of ocean acidification on juvenile red king crab (Paralithodes camtschaticus) and Tanner crab (Chionoecetes bairdi) growth, condition, calcification, and survival. PLoS ONE 8(4): e60959. doi:10.1371/journal.pone.0060959.)

Anyway, as long as the ANOVA matches the experimental design (as one of the other answers suggested; if have a blocked or repeated measures design and you're running a 1-way ANOVA you're treating samples that are not independent as if they were, i.e. pseudoreplicating) then you shouldn't try to find a way to lower your F value.

This is very helpful. In my study, I got an F value of 267 in a repeated measure one-way ANOVA and I have been wondering if it is normal. Thank you