Except for a couple of minor points:
Unlike the settling of your fictional chain of debt, in reality every time the cash changes hands the government in whatever country the transaction is made will want its cut in taxation (to provide, among other things, funds for future bail-outs). This very quickly reduces the amount of cash changing hands.
Furthermore, if any of the participants in your chain happen to want a currency other that the blighted Euro, they will have to pop to the bank and change the Euros into their chosen currency and those wanting Euros further along the chain will have to do likewise, again reducing the sum passed on.
Lastly, unlike your fictional chain of debt, somewhere along the line the cash will encounter a body which is either unable or unwilling to settle its debts (unable because they may have larger debts elsewhere outside the chain which they see as more pressing, or unwilling because they have a government which, like our previous one, actively encourages the accumulation of large debt).
Other than that, it works a treat!