When you look at these devices, you may be tempted to think... well they've got low-res displays, or not as fast, etc. Just remember the rise of the "Japanese transistor radio" (and associated songs) and what that led to.
The most famous of these is MediaTek, and wow are they expanding rapidly. I just added them to my Arm vs Intel Benchmarks page. While not yet top tier, it is clear they will be soon. (I would suggest a betting pool, who'll become important in mobile first? Intel or MediaTek... but it's pretty clear MediaTek's already won that bet.)
Allwinner introduces A23 dual-core processor - Liliputing
Video: Qualcomm goes after MediaTek’s octo-core chips again
The market share for these manufacturers is expanding rapidly. IDC "Others" category for 2Q13 tablet shipments was 38.8% and for Android phone shipments was 37.5%.
During last year's holiday season, you could commonly find "white box" tablets in Amazon's Best Seller lists.
One cheeky white box manufacturer routinely makes knock off's of Apple's (and others) phones, but has improved to the point that Android Authority wrote this article:
GooPhone - cheap ripoffs or competent competition?
White box tablet and phone announcements are so numerous and common, I won't try to list them here (see Android Authority or Liliputing if you want to see plenty). There are so many, they even merit their own specialized products:
Huawei UltraStick could bring 3G to low-cost tablets - Liliputing
But look at these phone announcements:
Alcatel OneTouch Scribe Pro to rival Samsung's Galaxy Note in size, note-taking ability - Liliputing
ZTE U988S smartphone to pack Tegra 4 chip, $320 price tag (in China) - Liliputing
AnandTech | Xiaomi Announces the MI3 - Tegra 4 and MSM8974AB inside
Xiaomi will likely become a well known name soon:
Xiaomi - Business Insider
Xiaomi Director Says Valuation Reaches $10 Billion After Funding - Businessweek
Xiaomi Beats Apple to Smart TV With $490 Set Coming Next Month - Bloomberg
What China’s Xiaomi can teach Apple, Google, and the Western tech world | VentureBeat
Smart phones are increasingly being dominated by these manufacturers:
India, China: Lava, Intex Surpassed The Sales of Samsung and Apple Inc. (NASDAQ:AAPL) | eFinance Hub
China's smartphone makers catching up with western rivals | Business | The Guardian
This chart shows how far behind Apple is in the global phone market