Stars at near break-up rotation rates Accretion discs are ubiquitous in astrophisics. As a direct corollary, they are important for the following question.

Consider the following model, representing one of the most simple models for accretion discs. A central object is a star (pre-MS, WD or NS, but not a BH) of mass $M$, surrounded by a thin flat disc of material, which continuosly feeds the star at a rate $\dot{M}$, such that $M/\dot{M}$ is much larger than thermal and dynamical timescale of the star (i.e. accretion rate is slow).

Everywhere in the accretion disc its local motion is nearly circular and nearly Keplerian. Therefore, at the interface of the star and the disc the disc will always tend to make the star rotate at nearly-Keplerian velocities. From the other hand, if the stellar outer parts were to rotate at nearly-Keplerian velocities, these parts would become gravitationaly detached from the star, which would have significant consequences for the stellar shape and structure. Surely, though, the process is going to be slow and the acquired angular momentum will be redistributed within the star.

Now the question: What will be happening to the star if it approaches nearly break-up velocities due to such a spin-up? This involves a few subquestions: How close the rotation rate can actually get to the critical one? If it can get close enough, how would the whole process look like? That is, what would happen in the short term to the star when the effects of rotation will start to affect its structure? What would happen to the star in the long term?

I would like to keep this problem as a purely hydrodynamical one. That is, assume, that the only laws involved are hydrodynamical and gravitational ones, with some constant accretion rate supported. In reality magnetic fields would also play an important role for some stars, and stellar winds could also possibly be important.

Examples of the decribed systems are numerous. It might concern cataclysmic variables, millisecond pulsars, pre-main sequence star in a protoplanetary disc, and many more.
I have not the qualification to answer the question in its whole but the question is interesting (I worked on Be Stars which are episodically surrounded by an decretion disk and which rotates at nearly critical velocities. The phenomenon in Be stars is different from accreting stars. The only consequences of subcritical velocity is a flattened envelope and modification of its inner structure and of the oscillations modes found in these stars
(if you have time and curiosity, a good example of flattened star with Keplerian rotating decretion disk is Achernar, a Be star observed using interferometry --> Have a look at Meilland et al. 2007: www.aanda.org/articles/aa/pdf/2007/10/aa4848-06.pdf)

Anyway...

I found this paper about critically-rotating accretors. May be you will find answers to your questions here or in its references (use nasa ads site for your query: http://adsabs.harvard.edu/).
http: //arxiv.org/pdf/1306.1348v2.pdf
It seems that in the introduction, there is some answers to your questions about reaching the critical velocity.

The accreted mass can increase the rotation rate until the star reaches the critical velocity.

It is said : "For a typical 6 + 3.6 M⊙ system, with initial period
Pinit = 2.5 days, in the absence of spin-down mechanisms, only
3 per cent (0.12 M⊙) of the total amount of matter transferred
by RLOF (more than 5 M⊙) is enough to spin the gainer up
to the critical rotation."

But we still don't know if the gainer can really reach the critical velocity. Some papers are dealing with break-up mechanims which does not allow the gainer to reach critical velocity: spin down by tides, magnetic breaking, limitation of the accretion angular momentum through interaction with the accretion disk, stopping of the accretion mechanism...

I am sure that you will find many papers on nasa ads that will give you answers to your questions.