The simplest version of Johansen's (1988) trace test for cointegration is based on the squared sample canonical correlations between a random walk and its own innovations. Onatski and Wang (2017) show that the empirical distribution of such squared canonical correlations weakly converges to the Wachter distribution as the sample size and the dimensionality of the random walk go to infinity proportionally. In this paper we prove that, in addition, the extreme squared correlations almost surely converge to the upper and lower boundaries of the support of the Wachter distribution. This result yields strong laws of large numbers for the averages of functions of the squared canonical correlations that may be discontinuous or unbounded outside the support of the Wachter distribution. In particular, we establish the a.s. limit of the scaled Johansen's trace statistic, which has a logarithmic singularity at unity. We use this limit to derive a previously unknown analytic expression for the Bartlett-type correction coefficient for Johansen's test in a high-dimensional environment.