5 years ago, India launched the Mangalyaan spacecraft to Mars, on a budget that raised eyebrows at the time. It was probably the Mars mission launched with the least powerful rocket; a feat only made possible by the lengthy orbit-raising maneuvers that made use of the Oberth effect.
A similar approach was used by the Japanese Nozomi also on its way to Mars, and the Israeli Beresheet on its way to the moon. This can also be used to achieve a trans-Venusian injection.
On the other hand, gravity assists using VEEGA trajectories or similar have been proven by the Galileo and Cassini spacecraft, and soon will be performed by JUICE and Clipper.
Recently, a very exciting proposal called Trident was put forward (see:What is the most fuel efficient way out of the Solar System? and How can Trident be so inexpensive? Will it orbit Triton or just do a (slow) flyby?). This involves VEEGA to achieve a flyby to Jupiter, and then on to Neptune and beyond.
Breakthrough Initiatives announced their intention to privately fund a very austere flyby mission to Enceladus that presumably would then escape the solar system, I recall they said something like less than $100m. The Asian nations mentioned before could consider performing a low-budget mission to the outer solar system.
Is it possible to combine both approaches to perform flybys to the outer planets and escape the solar system?? Launch it first as if it was a Venus mission which heavily relies on oberth trickery, then use Venus and Earth gravity assists to reach Jupiter, and from there beyond