While there is no objectively correct answer to this, I would argue that the biggest challenge to many developing countries is the state of their governments. Governments in developing countries can often be corrupt and/or unstable.
When governments are corrupt or unstable, it is very difficult for economic development to occur in any systematic way. Governments may give monopolies to companies because the companies bribe government officials or because they have ties of some sort with those officials. This is not efficient. If governments are unstable, it is very hard for development of any sort to occur. Investors will be very wary of putting any money into the economy. They will fear that the government might collapse or that it might suddenly decide to confiscate their assets.
These kinds of problems with the government are very hard to eradicate. Therefore, they are one of the biggest challenges facing many less developed countries.