I believe that the answer you are looking for isn't as easy as it may seem. The limitation is caused on the receiving end as well as the sending end. You may find an app that will allow you to send >160 char as a single message & that will look all well & good on your screen. But the message is still being sent in multiple 160 char chunks so if the recipient's app isn't programmed to piece those chunks back together then it's not going to work. Most likely, there is NO app that can/will do that since there is no way for it to know that the multiple chunks are supposed to be together in the first place.
The technology is known as Short Messaging Service and therefore designed to only send short messages in 160 character chunks. If you want a longer message then you can possibly opt to send it as a MMS message (I know chompSMS allows that) but sometimes that can be awkward for the recipient. Or, for long messages, send an email. It's a limitation of the technology, not the device or app.