Ok, I'm going to be devil's advocate and try to make a case for this seldom-used teacher. I've always loved the art here, and I think it may actually have a place in decks with a variety of porous ICE, or even where a little extra tax is desired. So many ICE lack ETR subs, and what I find particularly appealing about this guy is that he doesn't just give an ETR to the next piece of ICE, but "each" piece of ICE behind him. He's also just beyond Net-Ready Yog range, which is pretty nice. On another note, he can make bioroids cost either an extra credit or click.

With Sensei, there are three categories that ICE fall into (excluding ICE that already has a solid ETR):

1. Synergistic - ICE lacking an ETR altogether that benefit by having one, either by adding tax or stopping power.
2. Boosted - ICE that has an ETR, but it is unreliable, such as traces or psi games, which would benefit from the addition of a solid ETR.
3. Anti-Synergistic - ICE that is designed to work better without an ETR, that complements the ICE behind it (like Sensei, himself), or trashes itself before an additional subroutine could fire.

As of Data and Destiny, the following is all ICE (again, excluding those with a solid ETR) which interact with Sensei, by category:

CATEGORY 1 - SYNERGISTIC

Archangel; Architect; Assassin; Bandwidth (synergy much?); Builder; Bullfrog; Burke Bugs; Cortex Lock; Crick; Data Hound; Data Raven; Enforcer 1.0; Errand Boy; Excalibur (which feels pretty darn good if they don't have an AI); Gemini; Grim; Gutenberg; Hourglass; Hunter; Ichi 1.0 (who suddenly looks like an HB Archer); Ichi 2.0; Information Overload; Ireress; Janus 1.0; Komainu; Lancelot (when lacking Galahad); Lycan; Mamba; Matrix Analyzer; Merlin (when lacking Galahad); Minelayer (though it is slightly anti-synergistic, as the newly-placed ICE won't benefit from Sensei, without some Tenma Line shenanigans); Nebula; Neural Katana; NEXT Gold; Pachinko; Pup; RSVP (provided the first subroutine fires, but this is an odd case); Sagittarius; Salvage; Searchlight; Shadow; Sherlock 1.0; Shiro; Snoop; Swarm; Swordsman; Taurus; Tsurugi; Turnpike; Tyrant; Virgo; Woodcutter; Wotan (just cruel); Yagura; and Zed 1.0.

CATEGORY 2 - BOOSTED

Caduceus; Clairvoyant Monitor; Dracō; Flare; Muckraker; Negotiator; News Hound; Pop-up Window; Resistor; Shinobi; Snowflake; Tour Guide; Troll; Turing; Uroboros; Viper; and Wormhole.

CATEGORY 3 - ANTI-SYNERGISTIC

Chum; Cell Portal; Checkpoint; Data Mine; Howler; Inazuma; Gyri Labyrinth; Kitsune; Lab Dog; Hudson 1.0; It's a Trap!; Lockdown; Marker; Sensei; Special Offer; Susanoo-No-Mikoto; TL;DR; Universal Connectivity Fee; Wendigo; and Whirlpool.

OVERALL ASSESSMENT

While I don't think Sensei is at the cutting edge of competitive play, far from it, in fact, I do believe there are a number of significantly impactful niche uses for it. While it may sometimes be a simple tax, there are just as many occasions where that extra ETR can trigger some interesting or just plain mean effects (i.e. Bandwidth or Wotan).

Sensei's real problem is Cost vs Gain.

History Lesson

Waaaaay back the days of the Genesis cycle, when dinosaurs ruled the earth, Fantasy Flight Games thought adding subroutines was a powerful thing, forcing exorbitant costs upon runners to run. Probably because the card pool was so limited. Hell, criminals were using shit like Aurora and Peacock to break subs, and many a Noise player tried relentlessly to make Darwin work somehow. Hence, ice like Sensei, Woodcutter, and Salvage were designed to add subs with limitations and at high cost. Woodcutter must be rezzed first, then costs , per sub. Sensei has high install and rez costs relative to subs added.

Nowadays, runners either pay 1 per sub to break or use limited free abilities, such as Overmind counters or ID abilities like Quetzal or Adam/ABR. Or they don't give a shit how many subs your ice has. So, by today's standards, cards from the early packs that add subroutines are vastly underpowered, and I expect to see newer iterations of these cards in the next few years as these older packs get cycled out.

Sensei's Weaknesses

All positional ICE have the hidden cost of installation: to affect an ice following it, you must pay an additional 1 to install it in front of that ICE. In the case of positional ICE that affects the rest of the run (Sensei, RSVP, Wendigo), you must pay increasing amounts of in install costs to get increasing benefits.

Adding an ETR is weak because it doesn't anything nasty to your opponent - not like an alternate win condition like "Do all the net damage" would. Either they can't get into your server for a little bit until they find the right breakers (blocked run), or they have to pay more to get into a server (taxing).

Sensei's bad at blocking runs. If you're installing ice to end the run, just use ETR ice. If you're adding subs to the next ICE, I'd sooner use Marker, as being free-to-rez offsets the install cost. Or, seeing as this example assumes no decoder, Lotus Field would serve the same purpose without requiring other cards.

Sensei's pretty weak as taxing ice. The extra usually translates to an additional spent by the runner. The most you're hoping for is +1 cost to run per ice, which you paid to install Sensei. An overhead cost of 3 to rez Sensei means you're paying more than the runner until they've run around 3 times.

Exceptions to this would be breakers that use counters, such as D4v1d or Cerberus "Lady" H1: Lady would not enjoy Sensei-Eli-Eli. Also, it's good to point out that it costs the decoder staple Gordian Blade 4 to break, so the runner is likely to just let Sensei fire.

How to Use Sensei

To make Sensei playable, combine with abilities that lower install and/or rez costs, and install him on a deep server that your opponent will run often (HQ vs. Siphon recursion, for example). Refer to @slobad31's awesome review for synergistic ice.

Positional ICE is always difficult to make use of. See it too early, and it is useless without support cards to move ICE. See it late and it is nice, but did it really do its job then? Sensei is at least a piece of ICE that is not only difficult to break at five strength, but also adds subroutines to all other pieces of ICE after it. That adds the tax factor up really quickly. Given also that token breakers are becoming more popular (such as Cerberus "Lady" H1) an additional subroutine could run those kinds of breakers out faster, opening up scoring windows for the corp.

It is positional, and that is its greatest drawback for certain. It is an average of three of four credits to break though, which can be good - more so the more ICE that it is in front of. On average it probably is only going to be an additional two credits to break those subroutines it adds to the ICE behind it (assuming it averages out most often as the third ICE on a server), meaning its high tax to break will never be paid in favor of the smaller addition to further ICE.