Meven's answer explains that without inductive types, you cannot hope to recover a natural number object with proper induction principles. You can interpret this as saying that you lose some mathematical expressiveness.
However, in the Calculus of Constructions (CC) you can still define the Church integers
$N := \forall (X : \mathrm{Prop}) , X \to (X \to X) \to X$,
and even though they don't come with an induction principle, you can still define exactly the same functions as terms $N \to N$ as in CC + Inductives. So, in a way, you do not get any proof-theoretic strength out of inductives in an impredicative system.
Note that it is not true at all for predicative systems, such as MLTT, where inductive types are the main source of proof-theoretical strength.
Let's make this claim more precise: by CC, I mean the original system of Coquand, which can be expressed as a PTS with two sorts $\mathrm{Prop}$ and $\mathrm{Type}$, the first being impredicative.
For simplicity, I will only consider one inductive type, the integers. I use CC+Nat to mean CC extended with an inductive type $\mathbb{N} : \mathrm{Prop}$ with large elimination (and adequate computation rules):
$$
\frac{P : \mathbb{N} \to \mathrm{Prop} \quad t_0 : P\ 0 \quad P_S : \Pi\ n. P\ n\to P\ (S\ n) \quad n : \mathbb{N}}{\mathrm{natrec}(P, t_0, t_S, n) : P\ n}
$$
$$
\frac{K : \mathrm{Type} \quad P_0 : K \quad P_S : K\to K \quad n : \mathbb{N}}{\mathrm{Natrec}(K, P_0, P_S, n) : K}
$$
One can show that any term of type $N \to N$ (where $N$ still means the Church integers, not $\mathbb{N}$!) in CC+Nat corresponds to an integer function using the normalization theorem. I will first explain that any such function is provably recursive in higher-order Peano arithmetic without choice (PAω), and then I will explain that any function that is provably recursive in PAω can be expressed as a term in CC. Thus, CC+Nat and CC can define exactly the same functions on the Church integers.
From CC+Nat to PAω
This direction requires analyzing a normalization proof (this one doesn't treat large elimination, but we can follow Werner on this). Given any typing derivation in CC+Nat, our goal is to prove that the term is normalizing within PAω (we cannot do it uniformly for all derivations, as the proof strength of PAω is not sufficient).
First, we can show through simple syntactic consideration that in CC+Nat, terms can be stratified into kinds (inhabitants of $\mathrm{Type}$), predicates (inhabitants of a kind), and objects (inhabitants of a predicate of kind $\mathrm{Prop}$), and that nothing complex happens at the level of kinds — they can't be abstracted over, and they can be described by the following syntax:
$$
K ::= \mathrm{Prop}\ |\ \Pi (\alpha : K) . K\ |\ \Pi (x : T) . K\qquad \text{where $T$ has type Prop}
$$
The standard reducibility proof assigns a set-theoretic interpretation to each kind:
$$
\begin{split}
\mathcal{V}(\mathrm{Prop}) & = SAT \quad \text{The set of saturated sets of terms} \\
\mathcal{V}(\Pi (\alpha : K_1) . K_2) & = \mathcal{V}(K_1) \to \mathcal{V}(K_2)\\
\mathcal{V}(\Pi (\alpha : T) . K) & = \mathcal{V}(K_2)
\end{split}
$$
Unfortunately, we cannot reproduce this definition in PAω, because it "lives in kind ω+1".
But we are only interested in proving normalization for a specific derivation, and only finitely many kinds appear in that derivation — so we can restrict ourselves to a subtheory with bounded kinds, in which case the interpretation is definable in PAn for some n.
Then one proceeds to assign an element $[P]$ of $\mathcal{V}(K)$ to every predicate $P$ of kind $K$ by induction on the syntax, in the usual fashion (see this summary for instance. Integers and predicates defined by recursion are treated in Werner's thesis).
Finally, one shows by induction on the typing derivations that if $\vdash t : A$, then $t \in [P]$ and thus $t$ is normalizing.
We have that if $\vdash f : N \to N$ in CC+Nat, then PAω proves that for any Church integer $\overline{n}$, $f\ \overline{n}$ is normalizing (since it does not involve more kinds than $f$). After some simple considerations on normal forms, we can show that it normalizes to an integer $\overline{m}$, and thus we showed that the function coded by $f$ is recursive.
From PAω to CC
Girard's representation theorem shows that any function that is provably recursive in PAω can be expressed as a term in System Fω. The idea is to first use a double negation translation and "Friedman's trick" to reduce the problem to higher-order Heyting arithmetic HAω, and then translate HAω into Fω by erasing integers.
Finally, there is an obvious embedding of System Fω in CC.
Extensions
This proves that CC+Nat and CC can define the same functions on the Church integers. What about stronger systems?
Werner actually treats a full scheme of inductives with large elimination for small constructors (also known as "Impredicative-Set"). I would expect that the same technique carries through for the full scheme, but inductives are somewhat messy.
What about a universe hierarchy? We could hope that the full CIC (CC with a universe hierarchy and inductives) has the same proof-theoretic strength as CCω (CC with a universe hierarchy). However, I don't know any normalization proof for CIC — we cannot do the stratification by kind anymore, so it is not clear whether a similar technique works.
