I'm not sure which sort of examples of Lie algebras without the corresponding groups you have in mind, but here is a typical example from Physics.

Many physical systems can be described in a hamiltonian formalism.  The geometric data is usually a symplectic manifold $(M,\omega)$ and a smooth function $H: M \to \mathbb{R}$ called the *hamiltonian*.  If $f \in C^\infty(M)$ is any smooth function, let $X_f$ denote the vector field such that $i_{X_f}\omega = df$.  If $f,g \in C^\infty(M)$ we define their *Poisson bracket*
$$\lbrace f, g\rbrace = X_f(g).$$
It defines a Lie algebra structure on $C^\infty(M)$.  (In fact, a Poisson algebra structure once we take the commutative multiplication of functions into account.)

In this context one works with the Lie algebra $C^\infty(M)$ (or particular Lie subalgebras thereof) and not with the corresponding Lie groups, should they even exist.