My question is: Is there a way to make OS X partition recognized and I can boot into it again? I wouldn't want to reinstall OS X though.
Because the whole partition is intact.
In the old MBR system, you can only have 4 primary partitions (one of which can be an extended partition containing other logical partitions). In a pure UEFI system with GPT formatted drives (like a modern Mac), the old 4 partition limit doesnt apply. But with bootcamp things get a bit more fuzzy.
Bootcamp creates a hybrid partition table. So its a GPT disk, but the the old MBR based partition table is also used. Bootcamp will define a primary partition (of type/tag 0xEE) that encompasses the contents of all the GPT partitions (i.e. the EFI partition, OSX itself etc. Basically from sector 1 to the end of all your usual Mac/GPT stuff). The next entries in the MBR partition table will then define the rest as per usual, such as a windows drive on a primary partition, starting after the GPT content.
So the two comments above are both correct (you can define as many partitions as you want in the GPT portion, and OSX, or other EFI booted OS's will function fine). But bootcamp presents windows with a psuedo bios boot procedure, and sets up windows to appear that it is for all intents and purposes running on a traditional MBR formatted drive - with MBR limits. Now the option presented above, to create an extended partition within windows should in essence be fine. But there is a potential the MBR partition table will be updated, but the GPT wont - placing the two out of sync. Perhaps win10 handles this better than older versions, I'm not sure. If it occurs OSX wont see any extended/loglcal drives. This may or may not be a problem for you (and you can use tools like rEFit to resync the two, or if you are strong willed it can be done manually). There are other solutions to creating more partitions for windows with other cavets, but hopefully the information above will help you begin trouble shooting future issues.
Anyway back to the original question, I hope you have tried the "holding option key on chime" suggestion above, to either get into OSX or the Recovery partition. The suggestion to back up data before modifying anything further is a good one. If you cant get into either OSX or the Recovery partition by holding down the option key, you might need to create an OSX installer usb stick (on another mac) to boot into and repair things. If you can get into the Recovery but not into OSX, open up a terminal and type in "diskutil list", and tell us what response you get. Otherwise, depending on what works and what doesnt, hopefully someone in this thread can guide you further, if need be.