It is apparently difficult to use this adverb without redundancy, as illustrated by Bryan Garner in A Dictionary of Modern American Usage. As an example of its misuse, Garner presents the following excerpt from the New York times:

As the business becomes increasingly more competitive, do publishers care which books they publish or what shapes the manuscripts are in when they hit the press?

One remedy for the obvious redundancy, he suggests, is simply to avoid the troublesome word altogether:

As the business becomes more competitive, ...?

As a bolder alternative, he recommends that more be deleted instead:

As the business becomes increasingly competitive, ...?

But whereas his first proposal clearly conveys the intended meaning, Garner fails to realize that the second gives a very different result.

Let us suppose that the level of competition in the business begins to increase at some time t0 and continues to increase until some later time t1, and that the context of the question "do publishers care ...?" is the interval from t0 to t1. It is quite accurate to say that the business becomes more competitive throughout that interval. Note that more in this context means more than at the current time. Thus, even after the business has already been observed to become more competitive, there is no reason that it can't continue to become more competitive as the point of reference changes. Consequently, the first solution works.

On the other hand, if increasingly is understood to mean to an increasing degree, then it may be said that the business is increasingly competitive throughout the interval from t0 to t1. However, once it becomes increasingly competitive at time t0, since it is no longer in a position to become that which it already is, the business cannot be said to become increasingly competitive at any point in the interval later than t0.

Here is a solution that properly retains increasingly:

As the business continues to be increasingly competitive, ...?