Is there a non-parametric, two-way, continuous data ANOVA for replicated repeated measures? I have data of multiple factors. Let's call them: size (2 levels), geometry (2 levels) and time (242 levels but I can limit my focus to 3 levels, which are relevant).
I also have a measure (dependent variable) called "distance", which is continuous.
Description: I measured the distance of the cells to starting point in different geometries, different pore sizes over time. There are multiple measurements for each group. It looks like that the distribution of the distances of these groups are different from each other.
I would like to know, if distance can be predicted by size and geometry.
My data is not normally distributed, so I would like to apply a non-parametric test.
It would be great to include all time points to compare "curves" or time-course but if not possible, it is enough to do the test on 3 relevant time points.
I am using R.
I think I cannot use:
Friedman test, as it is for non-replicated data.
Kruskal Wallis, as it is one-way (last resort?).
OLR, as it is for ordinal data, not continuous.
 A: There are a couple of misunderstandings here.  The Kruskal-Wallis test is for non-replicated data; Friedman's test is for dependent data.  (You are right, though, that these are 'one-way' tests.)  In addition, continuous data are ordinal.  
Ordinal regression models (OLR / the proportional odds model) generalize tests like KW (see my answer here: What is the non-parametric equivalent of a two-way ANOVA that can include interactions?).  There are mixed-effects versions of OLR (see my answer here: Is there a two-way Friedman's test?).  There are also mixed-effects versions of proportional hazards models that could be used (cf., R's coxme).  These assume the random effects are normally distributed.  
With enough data (which you presumably have), you can count on the central limit theorem kicking in and use the GEE (to understand the contrast between mixed models and the GEE, see my answer here: Difference between generalized linear models & generalized linear mixed models).  
Another possibility is to aggregate the data into means that can be taken as independent and use standard methods to analyze the aggregates.  This will typically induce a loss of power, but can be a perfectly viable choice.  
You state your data are non-normal, but you don't say much about the nature or extent of the non-normality.  It may be fine to use a linear mixed model.  Alternatively, if the issue is that you can't have negative distances, but otherwise the data are somewhat normal-ish, they may be approximately Gamma, and there are models for that as well.  
