The potential of fly-by ping pong is pretty much unlimited, provided you have enough time to your disposal.
Given an initial transfer with a perihelion slightly lower than the orbit of Venus, a Venus flyby can increase the aphelion to a bit further out than the Earth's orbit.
On the following Earth flyby, the perihelion can be lowered, and you can just repeat this pattern until the eccentricity is high enough that the perihelion is inside the Sun.
In general: Given two planetary masses, repeated flybys can provide arbitrary eccentricity adjustments (in practice, you would likely want to speed up this scheme by involving Jupiter)
Cost: slightly more than 1 Venus transfer, ~3.5 km/s of $\Delta v$
As you probably know, just repeated Earth flybys will not have the same degree of eccentricity adjustment power, as the $v_{\infty}$ stays the same. However, the Earth system is not a single body, it's a two-body system including the Moon. If you instead of going for a full Venus transfer escape Earth with a small but non-zero $v_{\infty}$, repeated flybys of the Earth-Moon system can slowly increase this velocity, until you can reach Venus and execute the above scheme. This takes a very long time.
While going down the rabbit hole, an Earth escape isn't quite necessary either, as multiple Moon flybys will eventually give you an escape with some $v_{\infty}$.
Cost: a Moon transfer, ~3.12 km/s of $\Delta v$
For the mere 0.4 km/s of savings, one could question of the centuries of added flight time is worth it. On the other hand, this is the absolute minimum.
