Balance isn't the real issue here, stability is.
Pretty much any group could assume the third role simply by rivaling the other two in power and influence, but this structure will likely not last long before someone get's overthrown. In order to gain stability, there needs to be mutual dependence between the three powers. Although they are often at odds with each other, they each depend on the other two in some way. There is some animosity between them, but it is in each of their best interest to ultimately protect the others. The absence of any one would leave the other two weaker, not stronger. I have listed a few ideas below.
A powerful enough crime organization (mafia style) could certainly influence the government enough to stay in power. The don't want to overthrow the government as it benefits them and the government doesn't want to stop them due to internal corruption (bribery, extortion, threats, etc.). They would face more opposition from the church, but could also have corrupted enough of the clergy to maintain their power.
Wealthy noble class
The most elite families have enough social and financial power to influence both the church and the government. They want a say in what happens, but don't want to do the governing themselves, and the church and government both depend on their funding.
The common people
The lower classes do not have nearly enough power to directly oppose the church or government, but they can easily stop production and effectively the entire economy by refusing to work. (Effectively, a massive labor union.) They depend on the government and the church for protection, infrastructure, etc. and the church and government depend on the people being happy and productive. There can also exist a council or board that represents the people to the government. (Note: they don't make laws so they are not a legislature. They simply ensure fair treatment.)