Undeterred by history, let me say that I can certainly think of a field theory that doesn't have any local or global symmetries. For example, suppose I take a real field $$\phi$$ and add an interaction that looks like $$\phi^3$$. This action is not invariant under any discrete or continuous transformations that I can think of, but is a perfectly well-defined (albeit sick) field theory.

Firstly, it's clearly invariant with respect to translations: $$x^\mu\mapsto x^\mu+\epsilon$$, so the corresponding theory of physics will conserve energy and momentum. Secondly: that's one incredibly presumptuous statement to make! The fact that you can't think of one is hardly a reason to dismiss the existence!

To my knowledge, "field theory" is synonomous to a theory of physics which comes from extremising some Lagrangian. Suppose your Lagrangian density is:

$$\mathcal{L} = \partial^\mu\phi \partial_\mu \phi - V[\phi], \qquad [\phi] \sim (x,\phi,\partial\phi,\partial^2\phi, \ldots)$$

To find all the symmetries of such a Lagrangian, you can't seriously hope to just stare at it and find them out! And in general, it's not even enough to consider only symmetries that have some kind of geometric standing (called Lie-type symmetries). Such symmetries correspond to those with generators (i.e. the corresponding element of the associate Lie algebra) of the form

$$ V = \alpha^\mu(x) \frac{\partial}{\partial x^\mu} + \beta(x)\frac{\partial}{\partial\phi} \qquad (*)$$

and to find

*all* the symmetries of the Lagrangian, you need to solve the (large) system of overdetermined PDEs

$$ \mathrm{pr}^N(V) \Delta[\phi] = 0 \quad \textrm{for all solutions to }\Delta[\phi]=0$$

where $$\Delta[\phi]$$ represents the E-L equations associated with your field theory, N is the order of these equations and $$\mathrm{pr}^N$$ is the prolongation of the vector field $$V\in TM$$ to the N-th jet bundle $$\mathcal{J}^NTM$$. This is non-trivial in itself (computers can solve the system if you specify polynomial $$\alpha^\mu(x)$$, $$\beta(x)$$ of some given order), but to actually answer the full question we need to consider more than just Lie-type symmetries, but those called "generalised" symmetries, or Lie-Bäcklund symmetries! These have generators of the form:

$$V = \alpha [\phi]\frac{\partial}{\partial \phi} $$

for which there is no (finite-dimensional) geometric interpretation. Note the function $$\alpha$$ is now a function of $$[\phi]\sim (x,\phi,\partial\phi, ....)$$.

A

*deep* result of Noether tells us there is a one to one correspondence between generalised symmetries of the Lagrangian density and conservation laws for the system. So if you want to have a reasonable theories of physics (i.e. have some conservation laws!), then you'll need to have a few generalised symmetries at the very least. If you want your field theory have local symmetries (in the sense of gauge theories), then you want the Lie algebra spanned by the vector fields in (*) to be non-trivial, i.e. there exist non-commuting vector fields.