The Reynolds Number is defined as $Re = \frac{\rho v L}{\mu}$, where $\rho$ is the density, $v$ is the fluid velocity, $L$ is the characteristic length, and $\mu$ is the dynamic viscosity. By definition, inviscid flow implies that $\mu=0$. By this formula, this would make the reynolds number infinite and thus turbulent.
Is it possible to have laminar inviscid flow? Must inviscid flow always (necessarily) be turbulent?
No, inviscid flows are not necessarily turbulent. If there is nothing to "trip" the turbulence, then the flow will remain laminar. Features which could trip the turbulence include vibration, small temperature fluctuations, any geometric imperfections, velocity field imperfections, and other similar things.
For example, potential flow is a type of inviscid flow. Potential flow solutions are laminar solutions to the Navier-Stokes equations.
As another example, if care is taken to avoid vibration and other flow imperfections, it seems you can generate laminar pipe flows at arbitrarily high Reynolds numbers. Laminar pipe flows have been obtained at Reynolds numbers of about 100,000 under these conditions, which is far higher than the typical transition Reynolds number of about 2,000. From what I understand, there is no indication that 100,000 is any hard limit; you probably could get higher with better experimental setups.
How I think about it is this: Viscosity helps damp out flow imperfections, allowing laminar flows to occur with more imperfections. If you have a truly inviscid flow, it needs to be perfect to not trigger instabilities which lead to turbulence. If you were able to obtain an inviscid flow, you should expect turbulence due to the imperfections inherent in reality.
To answer a question you put in a comment in the other answer, yes, I do believe using a turbulence model for inviscid flow is prudent. For RANS, the Reynolds stress still will exist if the flow is inviscid, and for LES the same is true for the residual stress.
The simple answer
Reynolds number can be more simply defined as the ratio of Inertial forces to Viscous forces.
This is why when speed/flow rate increases fluids go towards/become turbulent as you're increasing the inertia of the fluid.
Based on this logic if you had inviscid (no viscous forces) flow then the inertial forces would have to dominate which should lead to Turbulent flow.
The fringe case
If you're asking if it's possible as a 'fringe case' well... then we're getting to the boundary between engineering and science. Inviscid flow in itself is just an assumption to simplify the NS equations.
AFAIK In real life there's nothing that is truly inviscid, just things that are almost inviscid. Thus as soon as you have some incredibly-close-to-zero viscosity value, then you should have some equally incredibly-close-to-zero flow value that allows laminar flow.
