I’ve heard that LaTeX is Turing complete. Are there any programs written in LaTeX

add the following to you preamble:

\newcommand{\newCommandName}{text to insert}

Then you can just use \newCommandName{} in the text

For more info on \newcommand, see e.g. wikibooks

Example:

\documentclass{article}
\newcommand\x{30}
\begin{document}
\x
\end{document}

Output:

30

The point of stating that a mathematical model is Turing Complete is to reveal the capability of the model to perform any calculation, given a sufficient amount of resources (i.e. infinite), not to show whether a specific implementation of a model does have those resources. Non-Turing complete models would not be able to handle a specific set of calculations, even with enough resources, something that reveals a difference in the way the two models operate, even when they have limited resources. Of course, to prove this property, you have to do have to assume that the models are able to use an infinite amount of resources, but this property of a model is relevant even when resources are limited.