Besides the other good answers for you to check, i'm going to offer another.
I'm interested in the modprobe timeout message you are getting along with dropping to the busybox prompt.
The modprobe command is used to load drivers into the kernel to control your hardware. I have been doing some checking on my system.
The error you are getting is (paraphrased)
The 2 numbers i have bolded are the vendor id and the device id of the hardware that modprobe is trying to load the driver for.
... timeout ... /sbin/modprobe pci:v00008086d00001c03sv00001025sd00000635......
It's your sata ahci driver for your hard drive controller.
That's interesting because it's dropping you to the initramfs prompt because it cannot find the root filing system on your hard drive and the sata ahci controller talks to the hard drive where the root filing system resides.
"), device: 1c03
("6 Series/C200 Series Chipset Family 6 port SATA AHCI Controller
A search for the driver on my laptop using the device ids returns this
So the achi driver gets loaded when modprobe is called with that long string of characters.
matthew-S206:/home/matthew/storage/linux_source/initramfs % grep -i "v00008086d00001c03" /lib/modules/$(uname -r)/modules.alias
alias pci:v00008086d00001C03sv*sd*bc*sc*i* ahci
This driver is part of the initramfs filing system as well as on your root partition.
The initial ram filing system (initramfs) is loaded at startup and is used to load drivers necessary to mount the root filing system.
Extracting my initamfs and searching for the ahci driver..
mkdir initramfs && cd $_ && zcat /boot/initrd.img-3.9.0-3-generic | cpio -i && find . -name "*ahci*"
My initramfs contains the ahci drivers and on my machine they are modprobed into the kernel. I am not sure if that happens during initramfs or after the root filing system is loaded though.
matthew-S206:/home/matthew/storage/linux_source % mkdir initramfs && cd $_ && zcat /boot/initrd.img-3.9.0-3-generic | cpio -i && find . -name "*ahci*"
I wondering if this is why you are being dropped to the initramfs prompt; initramfs cannot load the ahci drivers to mount your root filing system.
matthew-S206:/home/matthew % lsmod | grep ahci
ahci 25819 5
libahci 31898 1 ahci
Have you had a kernel update and can you boot into that older kernel from the grub menu ?
Can you manually modprobe the driver from the initramfs prompt and does that help if you continue the boot process ? Does you initramfs contain the ahci and libahci modules ?
This is speculation on my part as i have never had a modprobe timeout during initramfs but it may be worth exploring further.