An EAI implementation is not something to be undertaken lightly. You will touch every major system--possibly every system, period--in your organization. Businesspeople, technical staffers and managers will need to work together.
Done right, EAI can save you thousands of lines of code and many labor hours of management. It can also give you new insight into your business processes and enable you to quickly connect systems that might otherwise be unable to communicate. Done wrong, it can cost you hundreds of thousands of dollars and produce no benefits.
If you want to implement EAI, inventory all the system interactions in your company--you'll be surprised at how many there are--and identify the most likely EAI candidates. Perhaps you have systems that are not communicating reliably, or systems for which "the guru" is no longer with the company. Then there are systems so critical that to lose them would cost big bucks every hour they're down. These are all good candidates for the first phase of an EAI deployment.
Now pick a couple of low-threat interfaces and use them as learning experiences. Choose test systems where you have in-house experts in the source and target systems, so mistakes can be patched easily. Run a sample implementation with these low-risk interactions, and then start Phase 1 of the rollout.
Getting high-risk integrations into Phase 1 is a tactical choice: The biggest business benefit is going to come from stabilizing unreliable interfaces or replacing homegrown integration code that no longer has an owner. The process of integrating your enterprise will take an investment of many labor hours, so showing a positive result early on is important.