Thomson’s infinite lamp as a mathematical monster

Imagine that we have a lamp – it is switched off at its initial state and this state can be changed by pressing a button. Having been an hour to play with this lamp, we switch the lamp on after 30 minutes. Then after waiting for 15 minutes, we switch it off – and then we switch it on again exactly 7.5 minutes later – and so on. I think that the end of the story is self-evident: after one hour (and neglecting that it is physically impossible) we pressed the button infinitely many times.
But what will be the result? Will the lamp light? Or will not?
It seems to be an unanswerable question – after all, we can regard the switch off state as an “odd” and the switch on as an "even" number (or vice versa). The source of this problem is that only a natural number is either odd or even – but infinite is not a number in a traditional way.
But there is another analogy and it can help. Nobody knows PI’s exact value since it is an irrational number. What is more, according to our actual knowledge, its digits are randomly distributed. But if we would be able to compute all of its digits, would the last digit be an even number?
Perhaps it seems to be an acceptable answer that there is no a last digit of PI, so it is neither odd nor even. But PI is nothing more than the ratio of the circumference of a circle to its diameter and although we do not know exactly the numerical value of this ratio, it is a certain, existing value. Computing more and more digits of PI, we’ll know it more and more accurately – and computing it to the infinity, we’ll know it exactly.
Ad analogiam: if we press the button of the lamp infinitely many times, then the lamp will be necessarily either switched on or switched off – although we cannot predict the lamp’s state.
At this point we can distinguish to different types in math: random and compressible strings. The previous one means that we cannot find a representation of the given string which is shorter than the original one. Heads and tails is a good example for it: you won't know the result without tossing the coin in reality.
Or onecould mention the cellular automatons (CAs). The state of their cells depend on the neighboring cells’ states and a CA changes in discrete steps. The result is that although the system is absolute deterministic (certain starting configurations always results the same next phases), cellular automation is an incompressible process. We cannot compute the next phase without executing the program itself.
Opposite to these above mentioned examples, a compressible string can be regarded to be “regular” in a sense that if we know the rule, then we can find the nth digit without computing others.
Our lamp represents a totally different solution. We can compute its every stage and its algorithm is ridiculously simple, so it is compressible - except for its endpoint. We cannot answer whether the lamp is switched on at its final stage – unless we de facto pressed that button for infinitely many times.
I wonder whether there are other, strange categories – for example, who could imagine a string which is compressible only at its endpoint? Perhaps other mathematical monsters lurking somewhere.