The reason for connecting to many peers is for security. If you connect to only a few peers, or you design your peer-finding system poorly, an attacker might arrange things so that all of your connections were to computers that he controlled. Then he'd be able to hide published blocks from you, and to send blocks and transactions to you that don't reach the rest of the network. This would allow him to mount something sort of like a 51% attack, but with much less computing power.

In other words, the peer-finding algorithm is security-critical, so you shouldn't mess with it unless you know exactly what you're doing. The transaction and block relaying algorithms are, too, by the way; if a large fraction of the network stopped relaying because they wanted to be lightweight, then 125 connections might not be enough to stay safe anymore.