Following the official rules of Jenga, what is the highest possible level that can be placed?
51 levels. Achieving this requires 97 moves.

Let's start with a fresh tower, and let's try to minimize the number of blocks in the top levels. The rest of the tower will consist of one-block levels.
I shall denote the state by the number of blocks on the top level, followed by the number of blocks on the next level down, etc. So a fresh tower is 3,3,3,...
One can't remove a block from the uppermost complete level, so our attempt goes as follows:

1,3,2,... ⇒ 6 in top 3.
2,3,1,... ⇒ 6 in top 3.
3,3,1,... ⇒ 7 in top 3.

And from there it loops.
Jenga has 54 blocks, so the tallest tower can have at most 51 level.

1,3,2,&lt;48 levels with just 1&gt; after 97 moves.
2,3,1,&lt;48 levels with just 1&gt; after 98 moves.

From the above, we can also determine that the tallest tower with a complete top level will have 50 levels. This takes 96 moves.