First of all even if there is a possibility to get a Visa-on-arrival there is always the chance of the country to deny the entry of a person. China also grants itself this right. see e.g. German official hints for entering China To have a planning security apply for a Visa upfront. This reduces On-arrival-denial.
Even if you have a valid Visa a country can always deny the entry of a person in specific cases. China also grants itself this right.
When applying for the Visa you specifically sign the conditions for a granted Visa.
If you are already in China and try to get a longer valid stay this is really not easy to make it happen, see the comment and link from Blaszard.
A common remark about Visa for China and stays longer than 30 days
If you are still at home and plan to go to China and need to stay for a longer period than 30 days there are 3 ways, one good, one so-so and one bad:
Good Apply for a Visa for longer stay (e.g. work visa), tourist Visa to my knowledge does not work. If granted you can stay there, no need to worry here.
So-so Apply for multiple entry Visa. If you don't know a person in China you have to show valid flights and hotels for both stays.
- To use your stays do the following: Just stay the first stay, on day 30 you can go to Shengzhen train station, walk to Hongkong and re-enter the country to stay your second stay.
Bad A technique I absolutely do not recommend is to apply only for a 30 day entry and overstay it. I can absolutely not recomend this, it is quite costly (at this time it can cost if I remember correctly up to 15000 RMB) and this reduces your chance to re-enter China dramatically. In case you find you overstayed your Visa you should immediately go to the dedicated immigration center in the country to arrange things and pay the fine.
Source: Travelled to China quite a number of times. A close friend overstayed and had to pay the fine.
I wish everyone a good trip!