An operator is, by definition, a linear map from some given linear space $V$ to itself. And the definition of a linear map cannot, in general, be independent of the linear structure if the space it acts upon.
There are, however, algebraic objects that can be defined abstractly, and that they are always represented as bounded operators on some (actually many) Hilbert space(s). These algebraic objects are the ones forming involutive algebras, in particular C*-algebras. It is standard to consider the set of observables of a given physical system as forming an involutive algebra.
In non-relativstic quantum mechanics, the C* algebra of interest is fixed (up to *-isomorphisms), and it is uniquely (up to unitary isomorphisms) irreducibly represented in a Hilbert space. This irreducible representation is the so-called Schrödinger representation on $L^2(\mathbb{R}^d)$, where position and momentum are respectively the self-adjoint unbounded operators of multiplication and $1/i$-times differentiation with respect to the variable $x\in\mathbb{R}^d$. It is in this representation that the pseudodifferential calculus, that associates to (almost) any function (actually distribution) on the phase space $\mathrm{T}^*\mathbb{R}^d$ an operator from $\mathscr{S}(\mathbb{R}^d)$ to $\mathscr{S}'(\mathbb{R}^d)$ (that can, in some case, be restricted to a densely defined operator on $L^2(\mathbb{R}^d)$), is set.
Clarified all that, there are plenty of examples of physically relevant bounded self-adjoint operators in quantum mechanical systems. The foremost example is the spectral family $(P_\lambda)_{\lambda\in \mathbb{R}}$ of a given self-adjoint (possibly unbounded) observable $A$. Many people consider the spectral resolution much more physically relevant than the observable itself (there is a $1-1$ correspondence anyways), for $\lVert (P_{\lambda_1}-P_{\lambda_2})\psi\rVert^2$ is the probability that a measurement of the observable $A$ in the system in the state $\psi$ would yield a value in the interval $[\lambda_2,\lambda_1]\subseteq \mathbb{R}$ (assuming $\lambda_2 < \lambda_1$). The spectral family consists entirely of bounded (self-adjoint) operators (orthogonal projections, that each satisfy $P^2_{\lambda}=P_\lambda$).
