Range of feasible coefficients in an unlimited growth model
A solid limit: k must be greater than zero. Unless you're talking about some cannibalistic species or something that isn't suited to the model at all.

As long as the species is productive in the new environment: k is greater than 1. The population is probably growing or again you probably won't be using an exponential growth model. 

As mentioned before you would need to know the species for more information. But if you look at generation times and litter sizes:


Some bacterial generation times (from here) range from 10 to
2000 minutes (33 hours). So that is $k=2$ per generation time. Per
day you're looking at a lower bound of 2 per day and an upper bound
of $k=2^{14}=10^{43}$ per day.
Mice are something like 12 week generation time and a litter of 10
giving something like $k=10^{10}$ per year.
Elephants are one young every 25 years. So $k=16$ per century or so.


Of course this is all based on gross assumptions. But you're looking for guidelines for a unrealistic model so hopefully they'll do.