Magic money machine

Imagine you have a magic money machine. You can buy this for €1, and it spits out €1 each second for the upcoming 3 seconds. Buying this machine in bulk is a good deal, but if you can buy as many of these as you want each second, which strategy should you apply such that your reserve cash (so cash not invested in the magic money machine) grows as fast as possible.

First, let us define what fast-growing means. A series $u_i$ grows faster than another series $v_i$ if there exists a time $t$ such that the first series is always bigger than the second series from that point in time onwards. So
$u > v \Leftrightarrow \exists t \forall i > t: u_i > v_i$
and $u$ is fastest growing if and only if $\nexists v : u < v$
This is a partial ordering, so a strategy that has the fastest-growing reserve cash does not have to be unique.
Also note that we try to maximise the total reserve cash and not the total net worth.

If you try this yourself, you will soon find that such a strategy does not exist. We will prove this by proof of contradiction.

Imagine there is a strategy that does produce the fastest-growing reserve cash. There are two possibilities. One possibility is that the reserve cash is always zero. This obviously leads to a contradiction, because the strategy is beaten by a strategy that just keeps the single euro. So there has to be at least one point at which there is at least a single euro in reserve cash. This means that we could create a new strategy that is perfectly equivalent to the previous strategy, except that that single euro is invested in a magic money machine. As the magic money machine directly spits back out a euro, this will never make any future investments impossible. The balance does not chance anywhere, except for the following two seconds, were, one euro is added to the balance. Therefore, the balance is higher starting from those two additions, and the strategy is beaten, which leads to a contradiction.