You have to ask yourself what you want from the rubber. Do you want speed? Control? Spin?
In short, the only rubber that has more spin than H3 is TG2.
If you want spin, go for H3. If you want control, go for STN it has better feel and control. If you want speed, just go for an ESN rubber.
I might go even further. Some of the rubber proposed here are 10-20$. I would consider them somewhat disposable. 10$ is not that much, at least in the economy where I live, I spent twice as much per week driving to the training sessions. If you want to try a rubber in that price range why not just go for it. If in your economy it is not considered that ignore those two sentences.
If you want the best bang for your buck, Yinhe Jupiter 2. For the price of 10$, it is super hard to get a better tacky rubber. I would still take off factory tuning and add some seamoon, but that is me. It is much more consistent than commercial H3, cheaper or same price. Beware of the factory glue though, it is very sticky stuff.
If you just want to spend money once and just play with whatever best there is, I would go for Hurricane 3 Provincial. It is a staple for a reason.
I've settled with Battle 2 as many here did, but I did spend some money trying few other rubbers until I reached the place where I am at the moment. If you want to try stuff, there is nothing to be ashamed of. There is plenty of options to chose from as discussed in this topic. Ones you might be interested would be Sanwei Target, 729 Bloom Power and Battle 2, Xiom Vega China (sweet rubber, although demanding a bit), Yinhe Jupiter 2, the new more expensive Jupiters, and probably few others I can't type from the top of my head.
If you just want to play I suggest considering two options presented. They are very much critically acclaimed in the community and that is for a reason.