Your question seems to express the issue of bias mentioned in the Wikipedia article on effect sizes. It seems like you are concerned that you regression model will be over parametrized, which $R^2$ does not take into account - as you note, it just gets higher the more parameters you put into it. Using adjusted $R^2$ would seem to be ok to addresss this problem.
This feature of $R^2$ has been highlighted as a reason not to use $R^2$, normal or adjusted, as part of a model selection process. A possible solution would be to do model selection using an information criteria such as AIC or BIC to arrive at an appropriate regression model from which to take $R^2$ and calculate $f^2$.
Cohen's $f^2$ does not appear to be used that often, but has been recently recommended for use with mixed effects (aka hierarchical or multilevel) multiple regression By Selya et al. (2012): "A practical guide to calculating Cohen's $f^2$, a measure of local effect size, from PROC MIXED".
Reporting the effect size for particular parameters instead of just the whole model might be one general solution to the bias issue you raise. Selya et al. use Cohen's $f^2$ for comparing the effect sizes of different predictors within their model. They specify $f^2$ as
$$
f^2= \frac{{R_{AB}^2} - {R_{A}^2}}{1-{R_A^2}}
$$
Where ${R_{AB}^2}$ is a multiple regression model with all of their predictors, and ${R_{B}^2}$ is a model without the predictor (A) for which they want to calculate a "local" effect size. ${R_{AB}^2}$ and ${R_{B}^2}$ can be respecified for different focal parameters for comparing several different effect sizes from the same model. If your are worried that $R^2$ is inflated then this approach should equally penalize all calculations of the effect sizes for individual parameters.
A final thing that comes to mind is that you could calculate a confidence region around $R^2$ and even $f^2$. This can be done via bootstrapping; there might be packages in R that can do this automatically. I'm not sure if this is actually relevant to the general issue of bias in $R^2$ but it could at least express some degree of uncertainty in how how $R^2$ is.
In linear regression, $R^2$ can't decrease when you add a new regressor (another "x" variable), since the fit will be equal or better. It remains the same only if the new "x" is a linear combination of the previous ones you already had. But then is it better to have more variables? $R^2_{adj}$ takes this into account by penalizing models with more variables. In other words, the increase in $R^2$ (i.e. the improvement of the fitting) must be reasonably large for the inclusion of a new variable to cause an increase in $R^2_{adj}$. In conclusion, you should always use $R^2_{adj}$ when comparing models with a different number of regressors. In the case where your models have the same number, it doesn't really matter which one you use. So it is just better to use the adjusted one all the time.
Best Answer
$R^2$ measures goodness of fit. But it will not detect overfit because it will increase with any new predictor (unless it has already reached 1). Since adjusted $R^2$ can decrease it can show that some models overfit the data. I think other criteria like AIC and BIC which also do this may work better. So if the purpose is to measure goodness of fit $R^2$ is appropriate and expressions the percentage of variance explained by the model. If the purpose is to identify whether or not the model overfits the data, adjusted $R^2$ is more appropriate. The sample size enters only because when the number of parameters is large and the sample size is small the degree of overfitting will be more severe than under the same circumstances with a much larger sample size. For the larger sample size parameters that should be 0 will be estimated close to 0 and so will not hurt prediction as much as in a small sample where the coefficient could be inappropriately large.