How to compare two sides of patients with multiple measures per patient? Let's say we have $n$ patients and we did $l_i$ measures of some parameter at the left side of the patients and $k_i$ measures of the same parameter at the right side of them. So we got a table like this:
i    left side                  right side
1    0.1  0.2  0.9  0.5 -0.3    0.5  1.3  0.9 -0.6  1.3  1.3 -0.3
2    -0.4                       -1.2 -0.9 -1.1 -0.9  0.8
3    -0.7                       0.2
4    -0.1 -0.2  0.8             -0.2
5    -0.6 -0.3                  1.1 -1.3

$i$ denotes the number of the patient, i.e. each line refers to another patient. The numbers of measurements of each patient were chosen randomly.
We do not have any information about the distribution of the values.
The values in the table above are rounded, in reality we have more precise values, so rounding won't be an issue.
We want to know whether there is a significant difference between both sides.
Normally we could do a paired wilcoxon-test to compare both sides, but in this case I doubt we can do that because of having patients of which we have multiple measures.
 A: Here are a few options:
You could use a mixed effects model where patient is the random effect and right/left is the fixed effect.  This would assume a normal distribution and it is not obvious how much non-normality would affect the conclusions, some type of simulation would be helpful to asses your trust in the results, or perhaps an adjustment to make them more trustworthy.
You could do a Bayesian hierarchical model.  You would not need to assume normality, but some form of distributional assumption would be needed.
You could do a permutation test.  One way to do this would be to take the mean of all the right values minus the mean of all the left values, then randomly permute the values within each patient and find the difference again, repeat a bunch of times and see where your original difference falls in the distribution of permuted differences.  You could also fit the mixed effects model and use the permutation approach to do the test on the differences.
I personally would probably favor one of the permutation approaches.
A: How about a Wilcoxon signed rank test?
wilcox.test() with paired=TRUE will perform this.
