Let κ be a regular uncountable cardinal. We construct non-archimedean exponential logarithmic models (R, an, exp) of cardinality κ, for T_an, exp (the elementary theory of the reals with restricted analytic functions and exponentiation), which admit a family F of 2^κ exponentials of pairwise distinct growth rates. These model exhibits the following remarkable features:

1. For each exponential exp’ in F, (R, an, exp’) is a model of T_an, exp and is thus o- minimal.
2. All exponentials in F agree exactly on the convex valuation ring of R.

In particular, the germs of these exponentials are incompatible, in the sense that the structure (R, an, exp, exp’) is no longer o-minimal.