No, you can't do that with an op amp.
Op-amps require power supplies. Many opamps have dropouts just like your 7805, so the output cannot quite reach the power rails. Others, called rail-to-rail opamps, can operate over the entire supply voltage. They're just drawn as triangles in some textbooks because it gets tedious to draw the power supplies every time (same with logic symbols).
Even if you could power an op-amp to beyond the rails, it's not advisable as a replacement for a 7805. Most opamps are low current (<20mA), and suitable only as precision voltage references.
Your options are, from easiest and cheapest to most difficult and expensive.:
- Tolerate the 4.8V and associated droop as the battery discharges (4.8V is a nominal value, not exact/constant). Many simple circuits can run off of 4.8V just as well as 5V.
- Use to get a higher voltage battery pack or put more battery packs in series to raise the supply voltage.
- Use a low dropout regulator[LDO] to regulate to a slightly lower voltage (3.3V, for example). 78XX isn't the only series of regulators! Many LDOs are available with dropout voltages of less than 0.2V.
- Use a boost regulator.
The final option uses switching of current through an inductor to generate voltages above the input. You will probably need a boost/buck regulator, since your input is so close to your output.