What's the correct tool to compare means between very small groups? I'm not really used to VERY small samples, so I'm wondering what to do in the following situation:
I have 3 groups, A, B and C. Each of them consists of 3 observations. I want to test whether there is a difference, so actually a classical ANOVA problem, followed by some post-hoc analysis (I would like to use Tukey's test). However, I have only 3 observations per group, so no clue of normality. Moreover the variances are not more or less identical (actually they are quite different).
So, should I switch to a non-parametric alternative?
(Of course I know that the statistical analysis is not very meaningful with these sample sizes, the boxplots show it quite better, but a reviewer wanted to see some p-values as well and actually almost all my tests were significant...).
Thanks!

edit: So first of all, just as an example, for the first comparison the variances are 1 vs 6 vs 36 in the 3 groups (the means are 8,32,35). So I don't have a real feeling whether they are VERY different. For some other comparisons they are closer, of course we have to interpret them related to the mean.
So almost all of my ANOVAs are clearly significant (the differences in means are quite obvious), but I'm wondering whether I'm allowed to use these results. We don't want to do everything the reviewer wants, some things are just statistically non-sense here, but some analyses would be rather nice, and I like the results of the ANOVA and the TukeyHSD tests (here, e.g., A vs.B and A vs.C gets significant, as expected, but not B vs.C). It's completely clear when looking at the plots but sometimes it's nice to add a p-value...
I'd rather like using a basic, commonly used, approach (but it should be correct of course!) I'm a mathematician and not so experienced with real world applications yet,but what I learned is that for papers like these it's better to stick to the golden standard, if possible. Further I was wondering whether Kruskal-Wallis would make sense?
 A: Even if there are real differences among group population means, there is no
guarantee than an ANOVA with such modest numbers of replications, after post hoc comparisons, will
allow you to conclude which of the three group means is largest
and which is smallest.
If your variances seem obviously different, I suggest using
oneway.test in R (or similar procedure), which does not
assume equal variances in the three levels of the factor.
Consider the following fictitious data, very roughly modeled
after the description in the you Question:
set.seed(2022)
x1 = rnorm(3, 50, 4)
x2 = rnorm(3, 55, 6)
x3 = rnorm(3, 60, 5)
stripchart(list(x1,x2,x3), pch=19)


x = c(x1,x2,x3);  g = c(1,1,1,2,2,2,3,3,3)
oneway.test(x~g)

        One-way analysis of means 
        (not assuming equal variances)

data:  x and g
F = 5.2598, num df = 2.0000, denom df = 3.8527, p-value = 0.07918

There are significant differences at the 8% level, but not at the 5%
level, even though all observations in Group 3 are larger than
any observation in Group 2. An ordinary ANOVA would have had 6
denominator degrees of freedom--reduced to about 4 here on account
of differing group variances.
The following simulation shows that the power of oneway.test with such data is poor: only about 30% (more precisely $0.302 \pm 0.003).$
So, you will be doing ad hoc tests only occasionally.
set.seed(307)
pv = replicate(10^5, 
 oneway.test(c(rnorm(3,50,4),rnorm(3,55,6),rnorm(3,60,5))
              ~ g)$p.value)
mean(pv <= 0.05)
[1] 0.30242       # aprx power
2*sd(pv <= 0.05)/sqrt(10^5)
[1] 0.002904922   # aprx margin of simulation error

