  We consider a Gaussian multiple-access channel where the number of
transmitters grows with the blocklength $n$. For this setup, the maximum number
of bits that can be transmitted reliably per unit-energy is analyzed. We show
that if the number of users is of an order strictly above $n/\log n$, then the
users cannot achieve any positive rate per unit-energy. In contrast, if the
number of users is of order strictly below $n/\log n$, then each user can
achieve the single-user capacity per unit-energy $(\log e)/N_0$ (where $N_0/ 2$
is the noise power) by using an orthogonal access scheme such as time division
multiple access. We further demonstrate that orthogonal codebooks, which
achieve the capacity per unit-energy when the number of users is bounded, can
be strictly suboptimal.
