Is there an intuitive way of seeing why there are only finitely many irreducible representations? Let $G$ be a finite group.  A basic result in representation theory is that up to $\mathbb{C}[G]$-module isomorphism, there are only finitely many irreducible representations of $G$ over $\mathbb{C}$.  The way I'm familiar with proving this is to let $(\pi, W)$ be any irreducible representation of $G$ with character $\chi$, and let $(\phi, V)$ be the left regular representation of $G$ ($V$ is the vector space with formal basis $x_g : g \in G$, and the $\mathbb{C}[G]$-module structure is given by $gx_h = x_{gh}$) with character $\gamma$.  One can prove that "the number of times $W$ occurs in $V$" is equal to $$(\chi, \gamma) = \frac{1}{|G|} \sum\limits_{g \in G} \chi(g) \overline{\gamma(g)}$$ and one can then quickly argue that this is not zero.  
It follows that every irreducible representation of $G$ occurs as a direct summand in $V$, and from the limited uniqueness a representation has as a direct sum of irreducible subrepresentations, you can see that there are only finitely many irreducible representations up to isomorphism
However, is there a more intuitive way of seeing that there are only finitely many irreducible representations of $G$?  Without any character theory, I know that if $(\pi,W)$ is irreducible, then the dimension of $W$ must be $\leq$ the order of $G$, because if $0 \neq v \in W$, then the span of $gv : g \in G$ must be all of $W$.  I am having trouble coming up with any immediate results beyond that.
 A: Let $\rho$ be a representation of $G$ on the vector space $V$.  For any $v \in V$ and $l$ linear functional on $V$ consider the "matrix coefficient" $\phi_{l,v}$, a function from $G$ to $k$ defined as follows
$$\phi_{l,v}(g) = l (g v)$$
The equality $$\phi_{l,hv}(g) = l (g hv)= \phi_{l,v}(gh)$$
implies that the map from $V$ to $k$ valued functions on $G$, 
$$v\mapsto \phi_{l,v}(\cdot)$$
is a morphism of representations. Assume now that $V$ is irreducible and $l\ne 0$. Then we conclude that $V$ imbeds into the space of $k$ valued functions on $G$, that you may call $k[G]$. 
Thus, we see that every irreducible representation is a subrepresentation of $k[G]$. 
Assume now that $G$ is finite. Let us show that the sum of the dimensions of irreducible representations is $\le |G|$. Let $V_1$, $\ldots$, $V_N$ irreducible subrepresentations of $k[G]$ so that $\sum \dim V_i > |G|$. Let $i$ minimum so that the sum of $V_1$, $\ldots$, $V_i$ is not direct. Then $V_i \subset V_1 \oplus \cdots \oplus V_{i-1}$ and so it will be isomorphic to one of the $V_1$, $\ldots$, $V_{i-1}$. 
A: The proof of the Artin-Wedderburn theorem (together with Maschke's theorem) shows that $\mathbb{C}[G]$ is a finite direct product of matrix algebras $M_{d_i}(\mathbb{C})$, one for each irreducible representation of dimension $d_i$. In particular, it follows immediately, with no character theory, that the number of irreducible representations is at most $|G|$ (and even, again with no character theory, that $|G| = \sum_i d_i^2$). 
More generally, if $A$ is a finite-dimensional algebra over $\mathbb{C}$, simple modules of $A$ are the same as simple modules of the quotient $A/J(A)$ where $J(A)$ denotes the Jacobson radical. This ring is semisimple, and applying Artin-Wedderburn to it as above again implies that the number of simple modules is at most $\dim A$. 
You can consider these results as noncommutative analogues of the claim that a polynomial has finitely many roots, which you get by applying the above argument to commutative algebras of the form $\mathbb{C}[x]/f(x)$. The two results even overlap in the case of cyclic groups, which you get by setting $f(x) = x^n - 1$. 
Alternatively, if you're willing to believe that a representation is determined by its character, and that an irrep has dimension at most $|G|$, you can argue (without orthogonality) that there are finitely many irreducible characters by bounding the number of possible values an irreducible character in each dimension can have: namely, if $V$ is an $n$-dimensional irrep, then the character $\chi_V(g)$ is a sum of $\dim V$ roots of unity of order dividing the order of $g$. 
