You can, theoretically, build the equivalent of any IC from discrete components. (You may have to invent some yourself, like EEPROM cells...the demand for individual transistors with floating gates isn't high enough to get them into the Digi-Key catalog.) The performance will not be the same, though:
Size. Naturally, your component parts will be substantially larger than their IC equivalents.
Speed. Your larger components, and the longer leads used to connect them, will have substantially larger capacitances, inductances, and resistances than an IC design has to contend with. All these will contribute to reducing the operating speed of your design.
EMC. Your longer wires are more effective antennas for unintended communications between devices. Self-interference is even a worse problem, despite the increased distance between circuits.
Cost. A transistor from Digi-Key will cost a few cents, maybe under a cent in bulk, but individual transistors on an IC cost almost nothing.
Reliability. You'd think a large device would be stronger than a small one, but metallization held nearly rigid by silicon backing is substantially more durable than macroscopic solder connections, and the square-cube law weighs heavily in favor of the smaller implementation.
So, I'd say pick your fights...hand-build a 555, but buy your EEPROMS off the shelf.