[See Patterned Transcendental Numbers published in EJMT for more background.]
A theorem of Kurt Mahler generalizes a well known result of David Champernowne who showed that $0.1234567891011\cdots$ is a transcendental number. In particular, Mahler showed that given an increasing polynomial, $p(x)$, such that $p(n)$ is a non-negative integer for all non-negative integer inputs $n$, the number obtained by concatenating the outputs' digits (i.e., "$0.p(0)p(1)p(2)\cdots$") is transcendental.
It is not hard to extend Mahler's result as follows: Let $p(x)$ be any non-constant polynomial such that $p(n)$ is an integer for all integers $n \geq N$. Then concatenating the digits of the absolute values of the outputs for $p(n)$ to "$0.$", we get a transcendental number: $t=0.|p(N)|\,|p(N+1)|\,|p(N+2)|\, \cdots$ We call such numbers (polynomial) patterned transcendental numbers.
Note that if $t$ is any transcendental number, $a$ is a rational number, and $m$ is a non-zero rational number, then $m \cdot t+a$ is still a transcendental number. Given a polynomial, beginning input value, and rationals $a$ and $m$, the applet below computes some of the decimal expansion of our transcendental number $m \cdot t +a$. By choosing $a$ with some care, we can arbitrarily modify finitely many digits of our number. By choosing $m$ to be a power of 10, we can shift the decimal point to left or right.
The computation above is powered by SageMath. The Sage code is embedded in this webpage's html file. To view the code instruct your browser to show you this page's source. For example, in Chrome, right-click and choose "View page source".