Time series remains non-stationary, even after taking logarithm and second differences I have following data:
dat <- c(8.28, 8.47, 8.59, 8.48,8.50,8.06,8.31,8.09,8.26,7.93,7.77,7.57,7.53,7.75,8.39,8.50,8.61,8.79,8.78,8.80)
dat.TS <- ts(dat, start=1995, end=2014)


The adf.test() keeps suggesting that the data are non-stationary (p-value > 0.05), even if I take the logarithm and first or second differences. How is this possible and what is the reason for this?
 A: 
the data are non-stationary ... even if I take the logarithm and first or second differences. How is this possible and what is the reason for this?

While differencing may often make series near to stationary, the set of series that are rendered stationary by differencing are a tiny subset of the set of all series one might observe.
Here, for example are fifth differences of a series that are still clearly not stationary:

(In this case the reason is obvious enough, but it's easy to construct series that are not periodic when nevertheless would not be rendered stationary to any order of differencing.)
Note that if you have noise about a polynomial trend of order $k$ then it takes differences of order $k$ to remove it (a quadratic trend would require second differences). So if you have a series that looks like it could be regarded as a deterministic trend plus noise that doesn't look quadratic, then second differences may well not leave you with stationarity.
However, the noisier the series the greater the tendency of differencing to reduce the remaining trend to below detectability given the noise. Note that your series shows strong trend relative to the noisiness, so it might not be so surprising that second differences leave you with a series you could still tell from stationary.
A: It doesn't need logs. This data is well modeled with differences, an AR1 and 3 outliers.
[(1-B1)]Y(T) = +[X1(T)][(1-B1)][(-  .383)]        :PULSE          1999
                 +[X2(T)][(1-B1)][(+  .248)]        :PULSE          2008
                 +[X3(T)][(1-B1)][(+  .259)]        :PULSE          2002
                +     [(1-  .673B** 1)]**-1  [A(T)]
[]1
