The dice bonuses to Initiative are limited to 5d6, flat-out. That's why I said that they're a different sort of thing.
Let's give some examples:
Runner Ace has 3s in all his stats. Since we're only worrying about Reaction, Intuition, and Initiative I'll only use those stats in these examples. His stats are:
Reaction 3
Intuition 3
Initiative 6 + 1d6
Ace decides to get some Wired Reflexes installed, at rating 3.
His new stats are:
Reaction 3 (6)
Intuition 3
Initiative 3 (9) + 1d6 (4d6)
Now, Ace decides he's going to take some Kamikaze as well (he's feeling particularly suicidal). Now Ace's stats look like this:
Reaction 3 (6)
Intuition 3
Initiative 6 (9) + 1d6 (5d6)
Why didn't he get the extra d6 from Kamikaze? Because he cannot go higher than 5d6.
Once he's down from the Kamikaze, he decides to get Reaction Enhancers installed at rating 3. What are his stats now?
Reaction 3 (6/9)
Intuition 3
Initiative 6 (9/12) + 1d6 (4d6)
Most of the time, he's going to be running around without the bonuses of Reaction Enhancers. But when he has the Reaction Enhancers and the Wired Reflexes wirelessly enabled, they stack. If he does that, he ends up with a Reaction of 9, and an Initiative of 12 + 4d6.
Let's look at a different scenario entirely, the one that was presented by the OP.
Runner Bob is a physical adept with the Improved Reflexes power at rank 2. Let's say his stats look like this:
Reaction 3 (5)
Intuition 3
Initiative 6 (

+ 1d6 (3d6)
Now Bob decides to get a Suprathyroid Gland installed. Once the gland is installed, he gets a +1 to Agility, Body, Reaction, and Strength. This means his stats now look like this:
Reaction 3 (6)
Intuition 3
Initiative 6 (9) + 1d6 (3d6)
Let's try with just magical enhancements.
Runner Cleo has 3s in all stats. Cleo decides to cast Increase Reflexes on herself. She gets 4 hits, which gives her 4 extra Initiative and 2d6 extra Initiative Dice. Her stats now look like this:
Reaction 3
Intuition 3
Initiative 6 (10) + 1d6 (3d6)
The next turn, Cleo casts Increase Reaction on herself, getting 4 hits in the process. This means her stats now look like this:
Reaction 3 (7)
Intuition 3
Initiative 6 (14) + 1d6 (3d6)
The turn following, Cleo casts Increase Intuition and gets 6 hits. She only gets to keep 4 of those hits, because otherwise she's over her augmented maximum. Now her stats look like this:
Reaction 3 (7)
Intuition 3 (7)
Initiative 6 (18) + 1d6 (3d6)
Are you starting to see what I mean, and how the wording works?