There are multiple ways of looking at this and it depends on what questions you want to answer. As you have already realized, separate assessments on the same individuals are not the same as the same total number of assessments on separate individuals. The issue is that observations on the same individual will typically be correlated over time in some manner.
If the responses are sufficiently continuous, one reasonable approach would be to use a mixed effect repeated measures model (assuming that residuals turn out to to be reasonably close to normal). It is typically suggested that in the absence of definite knowledge on the correlation structure, it is sensible to assume an unstructured covariance matrix of the observations over time. Depending on whether the assessment at the beginning is prior to the start of the intervention of interest, this measurement could be a covariate (if it is prior to the start of the intervention) or be treated as a measurement (in either case). However, often it is problematic to treat it as an observation, if it was pre-intervention and some inclusion cut-off was applied to only include patients above or below certain values. If it is being used as a covariate, then one would typically also a assessment time by covariate interaction term (in addition to an asessment time model term).
The next big question is, which comparisons you actually want to do and what assumptions you are making. E.g. are you estimating a separate effect of the intervention at each time? Or are you assuming there is an effect of identical size at each time that is being estimated across all times? Depending on this your model might have further terms for intervention (e.g. dummy coded to 1=intervention group, 0=control group) and/or also time by intervention interaction terms (e.g. effect at beginning, effect at middle time, effect at end of study).
The denominator degrees of freedom (in part answering the question you were asking regarding the effective number of observations) would have to be estimated, one method that is supposed to have good operating characteristics is the Kenward-Rogers method implemented in many software packages (e.g. SAS).
A MANOVA would have at most 21 complete records (depending on whether missing values are always in the same individuals) and some use could be made of the incomplete records. E.g. the repeated measures model I described above would implicitly impute by the type of model I am describing, which would occur under a missing at random assumption, which may or may not be valid.
