By using WolframAlpha, I couldn't find any transcendental number without equal adjacent digits among the numbes $\tan(n)$, $\sin(n)$, $\cos(n)$, $\sec(n)$, $\cot(n)$, $\csc(n)$, $e^n$, and $ \log(n)$, where $n$ is an integer number.
How to find a transcendental number where no two adjacent decimal digits are equal?
Some results below
Start with Liouville's constant $0.11000100000000000000000100 \dots$ which is known to be transcendental and add $\frac{2}{99} = 0.02020202 \dots$. The resulting number is transcendental (because it differs from Liouville's constant by a rational) and has no identical adjacent digits.