= Solution
For a nonconstant <meromorphic> map, write $T_f(R)=m(R,f)+N(R;\infty)$ in the usual normalization, where $m=(2\pi)^{-1}\int\log^+|f(Re^{i\theta})|\,d\theta$. Let $N(R;a)$ count $a$-points with their local degrees, and let $\overline N(R;a)$ count each distinct point once, using the same integrated logarithmic weights. At infinity the local degree is the <pole> order. The <Nevanlinna deficiency> and <Nevanlinna ramification index> are
$$
\boxed{\delta_f(a)=1-\limsup_{R\to\infty}\frac{N(R;a)}{T_f(R)},\qquad
\theta_f(a)=\liminf_{R\to\infty}\frac{N(R;a)-\overline N(R;a)}{T_f(R)}.}
$$
The latter is an asymptotic excess-multiplicity index, not the integer local ramification degree. The <Nevanlinna first main theorem> gives $N(R;a)\le T_f(R)+O(1)$ and identifies deficiency with the lower limiting normalized proximity. The target $a$ ranges over the <Riemann sphere>, not just the real line.
The exponential is entire and has no <poles>. Consequently
$$
T_e(R)=\frac1{2\pi}\int_0^{2\pi}\max(0,R\cos\theta)\,d\theta
=\boxed{\frac R\pi}.
$$
Thus $c=1/\pi$ in this normalization.
To obtain the rational-composition growth, represent the nonconstant degree-$d$ sphere map by relatively prime homogeneous <polynomials> $(P,Q)$ of degree $d$. Their joint norm on the unit sphere in $\mathbb C^2$ has a positive minimum and finite maximum: simultaneous vanishing would contradict relative primeness on the projective line. Homogeneity therefore gives constants $0<A\le B<\infty$ with
$$
A\|(w,s)\|^d\le\|(P(w,s),Q(w,s))\|\le B\|(w,s)\|^d.
$$
Apply this to the entire nonvanishing lift $(e^z,1)$. The logarithmic joint norm differs from $d\log\|(e^z,1)\|$ by a bounded function. Its circular mean minus the value at zero is the spherical characteristic. It differs from the ordinary characteristic by $O(1)$: <Jensen's formula> applied to the entire denominator counts its zeros, while
$$
0\le\tfrac12\log(1+x^2)-\log^+x\le\tfrac12\log2
$$
controls the proximity replacement. If the denominator vanishes at zero, factor its zero and use its leading coefficient in <Jensen's formula>; the discrepancy is still a basepoint-dependent constant. This proves the <rational composition law for the Nevanlinna characteristic> here and gives
$$
\boxed{T_{U\circ e}(R)=\frac d\pi R+O(1)\sim\frac d\pi R.}
$$
If $U$ is constant, $d=0$ and its characteristic is bounded instead.
For a simultaneous nonzero deficiency and ramification example choose
$$
f(z)=e^z(e^z-1)^2,
$$
a degree-three <polynomial> composed with the exponential. Its characteristic is $3R/\pi+O(1)$. Its only zeros are $2\pi i n$, $n\in\mathbb Z$, all of <multiplicity> two. The number of distinct such points in a large disc is $R/\pi+O(1)$, so integration of their count gives
$$
\overline N(R;0)=R/\pi+O(\log R),\qquad N(R;0)=2R/\pi+O(\log R).
$$
The origin contribution is handled by its usual logarithmic counting term. Therefore
$$
\boxed{\delta_f(0)=\tfrac13,\qquad\theta_f(0)=\tfrac13.}
$$
This is <simultaneous deficiency and ramification for an exponential polynomial>: the missing exponential value contributes a deficit, while the attained value is attained with double <multiplicity>.
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