There aren't many WiFi-enabled alternatives in the same feature/price range.
EMW3165 might be the closest. It is similar to the ESP8266 but based on a mainstream STM32 ARM controller instead of Tensilica Xtensa. However it is not as ubiquitous as the ESP8266 so the price/module is also higher (around $8 per module), and the physical form factor is a bit harder to work with. Compared to the ESP8266, the EMW3165 has more flash, RAM, and more GPIO lines. Maybe a little bit faster vs ESP8266 at defaut speeds.
NL6621 is another competitor new to the market. It is also based on ARM, with a Cortex M3 MCU (160 MHz native speed). There's very little community info about this module. Pricing should be in the $3 to $5 range qty. 1 from sellers on AliExpress.
TI has a couple WiFI ARM MCU/modules (CC33xx series) but they are priced 10x more expensive than the ESP8266. The CC3200MOD starts at $25 qty.1 so they are obviously targeted for a different market.
Atmel also has a couple WiFi modules like the ATWINC1500. Like the first ESP8266 modules, the ATWINC1500 is not meant to be a standalone chip but rather to supplement a host MCU. They are almost as pricey as the TI modules so they're no longer popular for hobby use after the ESP8266 came out.