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.
Symmetries in this context are functions which Poisson commute with the hamiltonian, hence the centraliser of $H$ in $C^\infty(M)$. They define a Lie subalgebra of $C^\infty(M)$.