Actually, I remember that Kaiser did some experiments on this topic.

http://braukaiser.com/blog/blog/2013/03/19/access-to-air-and-its-effect-on-yeast-growth-in-starters/

That experiment pretty much correlates with my hypothesis.

This part of Kai's hypothesis is on the money.

"yeast growth in wort is limited by available oxygen for sterol production. That means that access to more O2 allows more cells to be grown."

However, the maximum number of viable cells that can be produced is limited by the maximum cell density per milliliter.

With that said, one thing that I am certain most readers will overlook in the experiment is the inoculation rate of 10.8 billion cells per liter. That inoculation rate is a fraction of the number of viable cells that are available from the average White Labs vial (it's an order of magnitude less than that of a relatively new White Labs vial).

To grow from 10.8 billion cells per liter to 250 billion cells per liter (which is above the normal maximum cell density of 200 million cells per milliliter), requires the culture to undergo log

_{2}(250 / 10.8 ) replication events.

Note: log

_{2} is the logarithm base-2 function, which can be performed on calculators that do not support it by taking the base-10 logarithm of 200 / 10 over the base-10 logarithm of 2 (i.e., log(250/10) / log(2)) or the natural (base-e) logarithm of 200 / 10 over the natural logarithm of 2 (i.e., ln(250/10) / ln(2)). This formula is the inverse of the basic exponential growth model after n replication events, which is cell_count_after_n_events = initial_cell_count x 2

^{n}.

number_cell_replication_events_kai_experiment = log(250/10.8 ) / log(2) = 4.53

To grow from 100 billion cells per liter to 250 billion cells per liter, the culture has to undergo log

_{2}(250 / 100) replication events.

number_cell_replication_events_fresh_white_labs_vial = log(250/100) / log(2) = 1.32

Stir plates appear to be yet another situation where the amateur brewing community has taken a technique that is rooted in science and generalized it to a point where the outcome from the technique no longer holds. In my humble opinion, little to nothing is gained by using a stir plate, stir bar, and a 2L to 5L Erlenmeyer flask at typical amateur brewer starter volumes. A stirred culture without continuous aeration offers no advantage over a well-shaken culture when propagating a White Labs vial, and a well-shaken culture has fewer possible sources of contamination.

A cheap and easy solution for shaken 1L starters is a 1-gallon glass jug with a sanitizable cap (be careful when pouring hot liquid in a glass jug because most jugs are made from untempered soda lime glass). For those who are not price sensitive, a 4L to 6L screw cap Erlenmeyer flask such as Corning model 4995-4L or 4995-6L can be a good investment if one takes care of one's gear, especially if purchased used. A 4L to 6L screw cap Erlenmeyer affords one the advantage of being able to boil one's culturing media in the flask. Larger starters will require a larger jug or flask (shoot for at least three times the starter volume).

Remember, the key to the well-shaken starter technique is using a vessel that is at least three times the size of the starter volume and shaking the starter until it is almost all foam after pitching. Periodically shaking the starter during the first six to twelve hours can be beneficial.