I have found this rather recent paper (2015) assessing that just 2 observations per variable are enough, as long as our interest is on the accuracy of estimated regression coefficients and standard errors (and on the empirical coverage of the resulting confidence intervals) and we use the adjusted $R^2$ and our interest is on having unbiased estimates:

(pdf)

Of course, as also acknowledged by the paper, (relative) unbiasedness does not necessarily imply having enough statistical power. However, power and sample size calculations are typically made by specifying the expected effects; in the case of multiple regression, this implies an hypothesis on the value of regression coefficients or on the correlation matrix between the regressors and the outcome must be made.

I have found this rather recent paper (2015) assessing that just 2 observations per variable are enough, as long as our interest is on the accuracy of estimated regression coefficients and standard errors (and on the empirical coverage of the resulting confidence intervals) and we use the adjusted $R^2$:

(pdf)

Of course, as also acknowledged by the paper, (relative) unbiasedness does not necessarily imply having enough statistical power. However, power and sample size calculations are typically made by specifying the expected effects; in the case of multiple regression, this implies an hypothesis on the value of regression coefficients or on the correlation matrix between the regressors and the outcome must be made. In practice, it depends on the strength of the correlation of regressors with the outcome and between themselves (obviously, the stronger the better for the correlation with the outcome, while things get worse with multicollinearity). For example, in the extreme case of two perfectly collinear variables, you can't perform the regression regardless of the number of observations, and even with only 2 covariates.