It is rarely accurate to extrapolate beyond the data, unless you know the
correct form of the equation.  Your verbal description does not sound like a
polynomial, which necessarily goes to +/- infinity as the independent
variable goes to +/- infinity.

Jerry

Consider using something like:

y = a * exp( b * (1-1/x)) rather than a polynomial.  This has only two
parameters and has the desired limited growth characteristic.

Since you say that the curve initially accelerates,
I visualize the curve as an "S" curve.
Maybe like this:
http://en.wikipedia.org/wiki/Generalised_logistic_curve
Here is the formula in Excel:
y=k_a+k_c/((1+k_t*EXP(-k_b*(x-2*k_m)))^(1/k_t))
where
y=  population
x=  time
k_a    lower asymptote
k_c    upper asymptote minus k_a
k_m    time of maximum growth
k_b    growth rate
k_t     asymmetry of max growth time

A simplified formula might look like this:
y=a/(b+EXP(-c*x))