Logic Fallacy in a single government purchase creates infinite GDP?
It seems that you misunderstood the concept of marginal propensity to consume. The marginal propensity is only valid for an infinitesimal increase in income. As soon as you move away from the original starting point where the marginal propensity was 1, there may be a different marginal propensity. The multiplier however assumes that the marginal propensity to consume remains the same at all income levels (since it is essentially an infinitely repeated application of the marginal propensity to consume). 

The above problem with such analyses is why economists switched to models with so-called microfoundations in which the firm and consumer behavior is explicitly modeled via profit/utility maximization. In a model with fixed resources it would quickly become clear that a marginal propensity to consume of 1 for all income levels would violate market clearing conditions.