Level up after maxing out character’s attributes

From the Oblivion Wiki:

Some magical effects can increase your skill above 100, but in general increases above 100 have no real effect. The only exceptions are Acrobatics and Athletics , which will continue to increase your character's jump height and run speed, respectively, up to a maximum value of 255.

As for attributes:

The current value of an attribute can be boosted past 100, and many, but not all, benefits of the attribute continue to increase for values greater than 100 (see Magical Effects on Attributes ). Most aspects of game play are based upon the current value of the attribute.

Fortifying an attribute temporarily raises that attribute's value by M points, for a period of D seconds. Fortify Attribute can increase an attribute above its base value and can even increase an attribute above the maximum value of 100. Increases in attributes past 100 have the same effect as below 100 with one exception: weapon damage. Luck's effect on other attributes of +5 Luck = +2 (Other Attribute) is not altered when Luck is above 100; however, as it effectively raises the level of other skills up to a cap of 100, Luck has a diminishing effect on attributes that exceed the cap, and ceases to have an effect on attributes that reach 100. See Oblivion:Luck and the other attribute pages for details of their effects on skills.

So while fortifying endurance past 100 will boost your health, fortifying strength past 100 will not boost your melee damage but it WILL continue to increase the maximum carry weight and fatigue.

You can make your own mini-mod for this. Since this sounds more like a question for GameDev, where we develop stuff in preference of using other people's tools ... let's explain how to go about it the "proper" way first:

1. If not already done so, install The Elder Scrolls Construction Set (version 1.2 is fine; you need the older 1.0 only for few specialised tasks which are broken in the latest version). Optionally, install the Oblivion Script Extender and the Construction Set Extender for significantly easier modding in general.
2. In the TES CS, load the main oblivion.esm by going to the menu item File -> Data, marking the tick to the left side of oblivion.esm, and clicking OK. The loading will take a while.
3. Open the list of quests by picking the menu item Character -> Quests...
4. The quest data window appears, with all the existing quests in the game (even those who aren't "real" quests, or are left-overs from the development) listed.
5. The next step depends on the quest.
   • Some have the level requirements listed right there, in the "quest conditions" part. In such a case, mark the offending line (it has a GetLevel check) with the mouse and hit the [DEL] key on your keyboard to remove it.
   • Other quests have the level check in the topics which start the quest. You'll have to work through the Topics tab and remove such conditions.
   • Some quests have the level condition (again, using player.getLevel) in the quests scripts. Click on the [...] button next to the script name to edit those.
   • Finally, both topic and quest stage result scripts can be used for such checks, but I don't believe any vanilla Oblivion quest does so.

For the lazy and in a hurry, there is a mod which does it for you: No Daedric Quests Level Locks.

(Version 1.2.0416)