Anova Analysis versus Time series analysis( hypothesis testing) I have some data that I believe follows a Repeated Measures design pattern.
But when I have shown this data to some, they tell me that one may want to approach this via time series analysis due to seeing a Seasonality Effect in the data.
So here is the nature of my data:
Imagine a Farmer in New Zealand grows oranges.
The farmer counts all the oranges grown per month, and over many years.
Question #1:
The question i wish to know, is to find out which years are considered to be the same amount of orange production and which 
 years are considered to be different, statistically significant.
(I know that one would use a Post-hoc test after ANOVA analysis.)
Question #2:
Can my question be answered via some kind of Time Series Analysis?
I have attached image of my data here:

 A: Yes, you need to use a time series method.  You need to take into account not just the seasonality, but the fact that the oranges grown in any one year not be independent of the number born in the year before or the year after.  ANOVA requires the observations to be independent; there are adjustments that can be made to take into account the fact that each year's observation is not as much new information as to be hoped for, but basically they all take you into the world of time series.  
You could get around the seasonality problem by just adding up all the months in each year - because your research question doesn't seem to mind about the distribution within the year, only comparing one year to another.  But this draws attention to another problem - why use the January to December year, rather than any other arbitrary choice of the 12 months that a year finishes in?  In fact, December appears to be bang in the middle of the oranges season, judging from your data, so it would surely make more sense to look at July - June years.  Noting that this still doesn't address the time series issue in the previous paragraph.
Further - even if you took July-June years, and the years weren't serially correlated, so ANOVA might be a method you could use, the statistical inference from it (F statistics and their p values) are only valid if you specified in advance, before looking at the data, which years you were testing for a significant difference.  By picking the most outlying years and then doing a test on them, you are strongly biasing it towards significance.  But as noted above, you shouldn't do this because of the time series angle anyway; I just wanted to point out this additional problem of invalidating inferences by choosing what you are testing based on the data.
So you have a challenge of "how do I identify the unusually low and high periods in a time series".  This probably deserves a question of its own; unfortunately there's no simple answer.  "outlier detection in time series" is probably the thing to google for, but the methods are all non-trivial.
