Are double pendulums eventually periodic?

Short answer: No. General trajectories of double pendulum are not periodic.

You need to distinguish between two aspects: the trajectory in the spatial coordinate system and the trajectory in phase space.

Your claim about $\gamma$ is about the first aspect and is thus false. It is perfectly okay for trajectories to intersect in the real space, and this doesn't mean the solution is periodic.

However, in the phase space it is forbidden for different trajectories to intersect (because of the uniqueness of the solution of ODEs given initial conditions). And if they do, you are correct that the dynamic is periodic. Indeed, notice that it may be that the mass travels through the same spatial point twice, but it can be with different velocities.

As @agemO suggested in a comment below, it is important to stress that although the solution is not periodic, it seem like it is getting close to there (which is probably what confuses you). Suppose for example that the mass starts from a point $(x,y)$ in the XY plane with velocity vector $(v_{x},v_{y})$. Then according to Poincare Recurrence Theorem, after some time the mass will travel as close as you want to that point with a very very similar velocity - but they are not guaranteed to be the same. In other words, the motion is as close as you want to be periodic, but it misses, and the resulting behavior is chaotic.

There is another very interesting theorem that should also be worth stating in this case. It is called Poincare Bendixson Theorem, and it states that a traped trajectory in 2D phase space must eventually repeat itself (given that the trapping region doesn't contain fixed points). But in this case the phase space is 4D and the theorem doesn't apply.

From wikipedia, the phase space of a dynamical system is defined as

... a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables.

For the double pendulum there are two position variables $\theta_1, \theta_2$ and two momentum variables $\dot \theta_1, \dot \theta_2$. So the phase space is 4 dimensional. If the system returns to the same point in its phase space its motion will repeat, because the state of the system is identical with its previous state so its subsequent motion is identical.

The 2 dimensional space with axes $\theta_1, \theta_2$ is not the phase space for this system. It is only a projection or "shadow" of the phase space.

Each point on the 2D plane $(\theta_1, \theta_2)$ corresponds to an infinite number of points in the 4D phase space, each with a different combination of $(\dot \theta_1, \dot \theta_2)$. The system can return to the same position $(\theta_1, \theta_2)$ with a different combination of momenta $(\dot \theta_1, \dot \theta_2)$ each time. The trajectory in the 2D plane $(\theta_1, \theta_2)$ can pass through the same point an infinite number of times without retracing the same path, because each of these intersections corresponds to a different state of the system.

So it is possible for the double pendulum to never return to the same point in its 4D phase space and therefore its motion can be non-periodic.

So your argument is not quite correct because of the existence of space-filling curves and fractals and the like; to take a simple example imagine a spiral over the whole plane $\mathbb R^2$, clearly not self-intersecting, and cut out the unit circle from it, then map it according to the Möbius transform $(x, y) \mapsto \left(\frac x{x^2 +y^2}, \frac y{x^2+y^2}\right).$ This transform maps the exterior of the unit circle into the circle, but it's totally invertible and cannot cause self-intersection. (It is also conformal and some other nice things that don't really matter here.) What you would get, if you look at this like a phase space, would be a harmonic oscillator with dissipation which never allows it to quite come to rest but always saps a tiny bit of energy from the thing. One could probably find that in the 4D phase space of a double pendulum there would be weird Hamiltonians which conserve energy and yet still do this. Trajectories need never be periodic precisely.

What you can prove is that for a gobsmackingly huge number of systems, the vast majority of the "nearby" paths to these exceptional ones will nonetheless return back to "nearby" whence they came, so that in general you will see a pattern of physics from $0 < t < \tau$ and then you will see an arbitrarily close copy of that from $T < t < T + \tau$ for some big $T$. This is known as the Poincaré recurrence theorem.